On some invariants of orbits in the flag variety under a symmetric subgroup

Sam Evens Sam Evens, Department of Mathematics, University of Notre Dame, IL, USA evens.1@nd.edu  and  Jiang-Hua Lu Jiang-Hua Lu, Department of Mathematics, Hong Kong University, Pokfulam Rd., Hong Kong jhlu@maths.hku.hk
(Date:: )
Abstract.

Let G𝐺Gitalic_G be a connected reductive algebraic group over an algebraically closed field 𝐤𝐤{\bf k}bold_k of characteristic not equal to 2222, let {\mathcal{B}}caligraphic_B be the variety of all Borel subgroups of G𝐺Gitalic_G, and let K𝐾Kitalic_K be a symmetric subgroup of G𝐺Gitalic_G. Fixing a closed K𝐾Kitalic_K-orbit in {\mathcal{B}}caligraphic_B, we associate to every K𝐾Kitalic_K-orbit on {\mathcal{B}}caligraphic_B some subsets of the Weyl group of G𝐺Gitalic_G, and we study them as invariants of the K𝐾Kitalic_K-orbits. When 𝐤=𝐤{\bf k}={\mathbb{C}}bold_k = blackboard_C, these invariants are used to determine when an orbit of a real form of G𝐺Gitalic_G and an orbit of a Borel subgroup of G𝐺Gitalic_G have non-empty intersection in {\mathcal{B}}caligraphic_B. We also characterize the invariants in terms of admissible paths in the set of K𝐾Kitalic_K-orbits in {\mathcal{B}}caligraphic_B.

1. Introduction

Let G𝐺Gitalic_G be a connected reductive algebraic group over an algebraically closed field 𝐤𝐤{\bf k}bold_k of characteristic not equal to 2222, and let {\mathcal{B}}caligraphic_B be the variety of Borel subgroups of G𝐺Gitalic_G with the conjugation action by G𝐺Gitalic_G. Let θ𝜃\thetaitalic_θ be an order 2222 automorphism of G𝐺Gitalic_G. The θ𝜃\thetaitalic_θ-fixed point subgroup K=Gθ𝐾superscript𝐺𝜃K=G^{\theta}italic_K = italic_G start_POSTSUPERSCRIPT italic_θ end_POSTSUPERSCRIPT acts on {\mathcal{B}}caligraphic_B with finitely many orbits [12, 13, 16].

Let V𝑉Vitalic_V be the finite set of K𝐾Kitalic_K-orbits in {\mathcal{B}}caligraphic_B, and for vV𝑣𝑉v\in Vitalic_v ∈ italic_V, let K(v)𝐾𝑣K(v)\subset{\mathcal{B}}italic_K ( italic_v ) ⊂ caligraphic_B be the corresponding K𝐾Kitalic_K-orbit in {\mathcal{B}}caligraphic_B. For a subset X𝑋Xitalic_X of {\mathcal{B}}caligraphic_B, let X¯¯𝑋\overline{X}¯ start_ARG italic_X end_ARG be the Zariski closure of X𝑋Xitalic_X in {\mathcal{B}}caligraphic_B. The Bruhat order on V𝑉Vitalic_V, denoted by \leq, is defined by

(1.1) v1v2  if  K(v1)K(v2)¯,v1,v2V.formulae-sequencesubscript𝑣1subscript𝑣2ifformulae-sequence𝐾subscript𝑣1¯𝐾subscript𝑣2subscript𝑣1subscript𝑣2𝑉v_{1}\leq v_{2}\hskip 14.454pt\mbox{if}\hskip 14.454ptK(v_{1})\subset\overline% {K(v_{2})},\hskip 14.454ptv_{1},v_{2}\in V.italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≤ italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT if italic_K ( italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ⊂ ¯ start_ARG italic_K ( italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) end_ARG , italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∈ italic_V .

When 𝐤=𝐤{\bf k}={\mathbb{C}}bold_k = blackboard_C, the geometry of the K𝐾Kitalic_K-orbits and their closures in {\mathcal{B}}caligraphic_B plays an important role in the representation theory of real forms of G𝐺Gitalic_G via the Beilinson-Bernstein correspondence (see [9]). The poset (V,)𝑉(V,\leq)( italic_V , ≤ ) and its application to representation theory have been the focus of extensive studies (see [1, 2, 3, 12, 13, 16, 17, 20]).

Let W𝑊Witalic_W be the canonical Weyl group of G𝐺Gitalic_G with the set S𝑆Sitalic_S of canonical generators, and let M(W,S)𝑀𝑊𝑆M(W,S)italic_M ( italic_W , italic_S ) be the corresponding monoid (see §§\lx@sectionsign§3.3 and §§\lx@sectionsign§3.6). Among the structures on V𝑉Vitalic_V are the monoidal action of M(W,S)𝑀𝑊𝑆M(W,S)italic_M ( italic_W , italic_S ) on V𝑉Vitalic_V (see §§\lx@sectionsign§3.6), the cross action of W𝑊Witalic_W on V𝑉Vitalic_V (see §§\lx@sectionsign§6.5), and the map ϕ:VW:italic-ϕ𝑉𝑊\phi:V\to Witalic_ϕ : italic_V → italic_W defined by Springer (see §§\lx@sectionsign§6.6). The programs from the Atlas of Lie groups (www.liegroups.org) facilitate explicit computations in examples of the cross action, the monoidal action, and the Springer map.

Let V0={vV:K(v)is closed}subscript𝑉0conditional-set𝑣𝑉𝐾𝑣is closedV_{0}=\{v\in V:K(v)\subset{\mathcal{B}}\;\mbox{is closed}\}italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = { italic_v ∈ italic_V : italic_K ( italic_v ) ⊂ caligraphic_B is closed }. In this paper, we associate to every v0V0subscript𝑣0subscript𝑉0v_{0}\in V_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and vV𝑣𝑉v\in Vitalic_v ∈ italic_V three subsets Zv0(v)Yv0(v)Wv0(v)subscript𝑍subscript𝑣0𝑣subscript𝑌subscript𝑣0𝑣subscript𝑊subscript𝑣0𝑣Z_{v_{0}}(v)\subset Y_{v_{0}}(v)\subset W_{v_{0}}(v)italic_Z start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) ⊂ italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) ⊂ italic_W start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) of the Weyl group W𝑊Witalic_W. The set Wv0(v)subscript𝑊subscript𝑣0𝑣W_{v_{0}}(v)italic_W start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) is defined both geometrically in terms of intersections of K(v)𝐾𝑣K(v)italic_K ( italic_v ) and Schubert varieties in {\mathcal{B}}caligraphic_B and combinatorially in terms of the Bruhat order on V𝑉Vitalic_V and the monoidal action of M(W,S)𝑀𝑊𝑆M(W,S)italic_M ( italic_W , italic_S ) on V𝑉Vitalic_V, and Yv0(v)subscript𝑌subscript𝑣0𝑣Y_{v_{0}}(v)italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) (resp. Zv0(v)subscript𝑍subscript𝑣0𝑣Z_{v_{0}}(v)italic_Z start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v )) is defined to be the set of minimal (resp. minimal length) elements in Wv0(v)subscript𝑊subscript𝑣0𝑣W_{v_{0}}(v)italic_W start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ).

One of our main results (see Theorem 2.2 and Corollary 2.3) says that for any v0V0subscript𝑣0subscript𝑉0v_{0}\in V_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, the subset Yv0(v)subscript𝑌subscript𝑣0𝑣Y_{v_{0}}(v)italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) (or Zv0(v)subscript𝑍subscript𝑣0𝑣Z_{v_{0}}(v)italic_Z start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v )) of W𝑊Witalic_W, together with the Springer invariant ϕ(v)Witalic-ϕ𝑣𝑊\phi(v)\in Witalic_ϕ ( italic_v ) ∈ italic_W, completely determine vV𝑣𝑉v\in Vitalic_v ∈ italic_V. This result generalizes [13, Theorem 5.2.2] of Richardson-Springer in the case when (G,θ)𝐺𝜃(G,\theta)( italic_G , italic_θ ) is of Hermitian symmetric type (see §§\lx@sectionsign§9.3). When V0subscript𝑉0V_{0}italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT contains a single element v0subscript𝑣0v_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, the set Yv0(v)subscript𝑌subscript𝑣0𝑣Y_{v_{0}}(v)italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) for any vV𝑣𝑉v\in Vitalic_v ∈ italic_V is a special case of an invariant for K(v)𝐾𝑣K(v)italic_K ( italic_v ) studied by Springer in [17].

When k=𝑘k={\mathbb{C}}italic_k = blackboard_C and K𝐾Kitalic_K is the complexification of a maximal compact subgroup of a real form G0subscript𝐺0G_{0}italic_G start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT of G𝐺Gitalic_G, we use the Matsuki duality between K𝐾Kitalic_K-orbits and G0subscript𝐺0G_{0}italic_G start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT-orbits in {\mathcal{B}}caligraphic_B to determine when a G0subscript𝐺0G_{0}italic_G start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT-orbit and a Bruhat cell in {\mathcal{B}}caligraphic_B have non-empty intersection. The problem of determining when two such orbits intersect comes from Poisson geometry and served as the main motivation for this paper. See §§\lx@sectionsign§2.3 and §§\lx@sectionsign§5.

We introduce the notion of admissible paths from v0V0subscript𝑣0subscript𝑉0v_{0}\in V_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT to vV𝑣𝑉v\in Vitalic_v ∈ italic_V by allowing only certain types of cross and monoidal actions by simple generators of W𝑊Witalic_W. We then characterize elements in Yv0(v)subscript𝑌subscript𝑣0𝑣Y_{v_{0}}(v)italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) (resp. Zv0(v)subscript𝑍subscript𝑣0𝑣Z_{v_{0}}(v)italic_Z start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v )) as products of the simple generators in minimal (resp. shortest) admissible paths from v0subscript𝑣0v_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT to v𝑣vitalic_v. See §§\lx@sectionsign§2.5 and §§\lx@sectionsign§8.

The technical part of the paper is an analysis in §§\lx@sectionsign§7 on the sets Yv0(v)subscript𝑌subscript𝑣0𝑣Y_{v_{0}}(v)italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) in relation to simple roots and their types relative to v𝑣vitalic_v. Various examples are computed in §§\lx@sectionsign§9.

Precise statements of the main results in the paper are given in §§\lx@sectionsign§2.

Most of the structures on V𝑉Vitalic_V are defined in [12, 13, 16] using a standard pair, i.e., a pair (B,H)𝐵𝐻(B,H)( italic_B , italic_H ), where B𝐵B\in{\mathcal{B}}italic_B ∈ caligraphic_B and HB𝐻𝐵H\subset Bitalic_H ⊂ italic_B a maximal torus of G𝐺Gitalic_G such that θ(B)=B𝜃𝐵𝐵\theta(B)=Bitalic_θ ( italic_B ) = italic_B and θ(H)=H𝜃𝐻𝐻\theta(H)=Hitalic_θ ( italic_H ) = italic_H. However, to formulate the sets Zv0(v)Yv0(v)Wv0(v)subscript𝑍subscript𝑣0𝑣subscript𝑌subscript𝑣0𝑣subscript𝑊subscript𝑣0𝑣Z_{v_{0}}(v)\subset Y_{v_{0}}(v)\subset W_{v_{0}}(v)italic_Z start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) ⊂ italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) ⊂ italic_W start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ), it is crucial that these structures can be defined independently of the standard pairs. Following [13, §§\lx@sectionsign§1.7], we review in §§\lx@sectionsign§3 and §§\lx@sectionsign§6 the canonical definitions of all the structures on V𝑉Vitalic_V needed in the paper.

Acknowledgments

We would like to thank Xuhua He and George McNinch for helpful discussions. We thank the organizers of the Atlas of Lie Groups meeting in Utah in July 2010, which helped the development of this paper, and also thank Jeff Adams, Peter Trapa, and David Vogan for very useful discussions. We are especially grateful to Scott Crofts, who wrote a program allowing us to compute the invariants Zv0(v)subscript𝑍subscript𝑣0𝑣Z_{v_{0}}(v)italic_Z start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) and Yv0(v)subscript𝑌subscript𝑣0𝑣Y_{v_{0}}(v)italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) explicitly in examples. The work described in this paper by the first author was partially supported by NSA grants H98230-08-0023 and H98230-11-1-0151, and that by the second author by the RGC grants HKU 7034/05P and 7037/07P of the Hong Kong SAR, China.

2. Statements of Results

2.1. Geometric and combinatorial definitions of Wv0(v)Wsubscript𝑊subscript𝑣0𝑣𝑊W_{v_{0}}(v)\subset Witalic_W start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) ⊂ italic_W

For v0V0subscript𝑣0subscript𝑉0v_{0}\in V_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and vV𝑣𝑉v\in Vitalic_v ∈ italic_V, we define Wv0(v)Wsubscript𝑊subscript𝑣0𝑣𝑊W_{v_{0}}(v)\subset Witalic_W start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) ⊂ italic_W by

(2.1) Wv0(v)={wW:K(v)B(w)¯},subscript𝑊subscript𝑣0𝑣conditional-set𝑤𝑊𝐾𝑣¯𝐵𝑤W_{v_{0}}(v)=\{w\in W:K(v)\cap\overline{B(w)}\neq\emptyset\},italic_W start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) = { italic_w ∈ italic_W : italic_K ( italic_v ) ∩ ¯ start_ARG italic_B ( italic_w ) end_ARG ≠ ∅ } ,

where B𝐵Bitalic_B is any Borel subgroup of G𝐺Gitalic_G contained in K(v0)𝐾subscript𝑣0K(v_{0})italic_K ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ), and for wW𝑤𝑊w\in Witalic_w ∈ italic_W, B(w)𝐵𝑤B(w)italic_B ( italic_w ) is the corresponding B𝐵Bitalic_B-orbit in {\mathcal{B}}caligraphic_B (see §§\lx@sectionsign§3.5). The set Wv0(v)subscript𝑊subscript𝑣0𝑣W_{v_{0}}(v)italic_W start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) depends only on v0subscript𝑣0v_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and v𝑣vitalic_v and not on the choice of BK(v0)𝐵𝐾subscript𝑣0B\in K(v_{0})italic_B ∈ italic_K ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) (see Lemma 4.1).

Denote the monoidal action of M(W,S)={m(w):wW}𝑀𝑊𝑆conditional-set𝑚𝑤𝑤𝑊M(W,S)=\{m(w):w\in W\}italic_M ( italic_W , italic_S ) = { italic_m ( italic_w ) : italic_w ∈ italic_W } on V𝑉Vitalic_V by m(w)v𝑚𝑤𝑣m(w)\!\cdot\!vitalic_m ( italic_w ) ⋅ italic_v for wW𝑤𝑊w\in Witalic_w ∈ italic_W and vV𝑣𝑉v\in Vitalic_v ∈ italic_V (see §§\lx@sectionsign§3.6). Our Lemma 4.2 gives the following combinatorial description of the set Wv0(v)subscript𝑊subscript𝑣0𝑣W_{v_{0}}(v)italic_W start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) for v0Vsubscript𝑣0𝑉v_{0}\in Vitalic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ italic_V and vV𝑣𝑉v\in Vitalic_v ∈ italic_V:

(2.2) Wv0(v)={wW:vm(w)v0}W,subscript𝑊subscript𝑣0𝑣conditional-set𝑤𝑊𝑣𝑚𝑤subscript𝑣0𝑊W_{v_{0}}(v)=\{w\in W:\;v\leq m(w)\!\cdot\!v_{0}\}\subset W,italic_W start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) = { italic_w ∈ italic_W : italic_v ≤ italic_m ( italic_w ) ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT } ⊂ italic_W ,

where recall that \leq is the Bruhat order on V𝑉Vitalic_V defined in (1.1).

2.2. The subsets Zv0(v)Yv0(v)subscript𝑍subscript𝑣0𝑣subscript𝑌subscript𝑣0𝑣Z_{v_{0}}(v)\subset Y_{v_{0}}(v)italic_Z start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) ⊂ italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) of Wv0(v)subscript𝑊subscript𝑣0𝑣W_{v_{0}}(v)italic_W start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v )

Let l:W:𝑙𝑊l:W\to{\mathbb{N}}italic_l : italic_W → blackboard_N and \leq be respectively the length function and the Bruhat order on W𝑊Witalic_W as a Coxeter group (see §§\lx@sectionsign§3.3 and §§\lx@sectionsign§3.4). Let W1subscript𝑊1W_{1}italic_W start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT be a subset of W𝑊Witalic_W. An element wW1𝑤subscript𝑊1w\in W_{1}italic_w ∈ italic_W start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT is said to be minimal if for any w1W1subscript𝑤1subscript𝑊1w_{1}\in W_{1}italic_w start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∈ italic_W start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, w1wsubscript𝑤1𝑤w_{1}\leq witalic_w start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≤ italic_w implies that w1=wsubscript𝑤1𝑤w_{1}=witalic_w start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_w. The set of all minimal elements in W1subscript𝑊1W_{1}italic_W start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT will be denoted by min(W1)subscript𝑊1\min(W_{1})roman_min ( italic_W start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ). An element wW1𝑤subscript𝑊1w\in W_{1}italic_w ∈ italic_W start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT is said to have minimal length if l(w)l(w1)𝑙𝑤𝑙subscript𝑤1l(w)\leq l(w_{1})italic_l ( italic_w ) ≤ italic_l ( italic_w start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) for every w1W1subscript𝑤1subscript𝑊1w_{1}\in W_{1}italic_w start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∈ italic_W start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT. The set of all minimal length elements in W1subscript𝑊1W_{1}italic_W start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT is denoted by minl(W1)subscript𝑙subscript𝑊1\min_{l}(W_{1})roman_min start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ( italic_W start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ).

For v0V0subscript𝑣0subscript𝑉0v_{0}\in V_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and vV𝑣𝑉v\in Vitalic_v ∈ italic_V, define

(2.3) Yv0(v)subscript𝑌subscript𝑣0𝑣\displaystyle Y_{v_{0}}(v)italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) =min(Wv0(v))Wv0(v),absentsubscript𝑊subscript𝑣0𝑣subscript𝑊subscript𝑣0𝑣\displaystyle=\min(W_{v_{0}}(v))\subset W_{v_{0}}(v),= roman_min ( italic_W start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) ) ⊂ italic_W start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) ,
(2.4) Zv0(v)subscript𝑍subscript𝑣0𝑣\displaystyle Z_{v_{0}}(v)italic_Z start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) =minl(Wv0(v))=minl(Yv0(v))Yv0(v).absentsubscriptmin𝑙subscript𝑊subscript𝑣0𝑣subscriptmin𝑙subscript𝑌subscript𝑣0𝑣subscript𝑌subscript𝑣0𝑣\displaystyle={\rm min}_{l}(W_{v_{0}}(v))={\rm min}_{l}(Y_{v_{0}}(v))\subset Y% _{v_{0}}(v).= roman_min start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ( italic_W start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) ) = roman_min start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ( italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) ) ⊂ italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) .

We prove in Lemma 4.3 that for any v0Vsubscript𝑣0𝑉v_{0}\in Vitalic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ italic_V and vV𝑣𝑉v\in Vitalic_v ∈ italic_V, Wv0(v)subscript𝑊subscript𝑣0𝑣W_{v_{0}}(v)italic_W start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) is determined by its subset Yv0(v)subscript𝑌subscript𝑣0𝑣Y_{v_{0}}(v)italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) in the sense that for any wW𝑤𝑊w\in Witalic_w ∈ italic_W,

(2.5) wWv0(v)  iff  ywfor someyYv0(v).formulae-sequence𝑤subscript𝑊subscript𝑣0𝑣iff𝑦𝑤for some𝑦subscript𝑌subscript𝑣0𝑣w\in W_{v_{0}}(v)\hskip 14.454pt\mbox{iff}\hskip 14.454pty\leq w\;\;\mbox{for % some}\;y\in Y_{v_{0}}(v).italic_w ∈ italic_W start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) iff italic_y ≤ italic_w for some italic_y ∈ italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) .

Thus both Zv0(v)subscript𝑍subscript𝑣0𝑣Z_{v_{0}}(v)italic_Z start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) and Wv0(v)subscript𝑊subscript𝑣0𝑣W_{v_{0}}(v)italic_W start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) are determined by Yv0(v)subscript𝑌subscript𝑣0𝑣Y_{v_{0}}(v)italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ). The example in §§\lx@sectionsign§9.4 shows that Zv0(v)subscript𝑍subscript𝑣0𝑣Z_{v_{0}}(v)italic_Z start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) may be a proper subset of Yv0(v)subscript𝑌subscript𝑣0𝑣Y_{v_{0}}(v)italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ).

Using the programs available at the Atlas of Lie groups, Scott Crofts has written a program that allows one to compute the sets Yv0(v)subscript𝑌subscript𝑣0𝑣Y_{v_{0}}(v)italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) explicitly in examples.

2.3. Intersections of real group orbits and Bruhat cells in {\mathcal{B}}caligraphic_B

Assume now that 𝐤=𝐤{\bf k}={\mathbb{C}}bold_k = blackboard_C and that K𝐾Kitalic_K is the complexification of a maximal compact subgroup of a real form G0subscript𝐺0G_{0}italic_G start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT of G𝐺Gitalic_G. For vV𝑣𝑉v\in Vitalic_v ∈ italic_V, let G0(v)subscript𝐺0𝑣G_{0}(v)\subset{\mathcal{B}}italic_G start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_v ) ⊂ caligraphic_B be the G0subscript𝐺0G_{0}italic_G start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT-orbit in {\mathcal{B}}caligraphic_B that is dual to K(v)𝐾𝑣K(v)\subset{\mathcal{B}}italic_K ( italic_v ) ⊂ caligraphic_B under the Matsuki duality [10] between G0subscript𝐺0G_{0}italic_G start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT-orbits and K𝐾Kitalic_K-orbits in {\mathcal{B}}caligraphic_B. The following Theorem 2.1 is the first main result of this paper.

Theorem 2.1.

Let v0V0subscript𝑣0subscript𝑉0v_{0}\in V_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and let BK(v0)𝐵𝐾subscript𝑣0B\in K(v_{0})italic_B ∈ italic_K ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ). Then for vV𝑣𝑉v\in Vitalic_v ∈ italic_V and wW𝑤𝑊w\in Witalic_w ∈ italic_W,

G0(v)B(w)  𝑖𝑓𝑓  wWv0(v).formulae-sequencesubscript𝐺0𝑣𝐵𝑤𝑖𝑓𝑓𝑤subscript𝑊subscript𝑣0𝑣G_{0}(v)\cap B(w)\neq\emptyset\hskip 14.454pt\mbox{iff}\hskip 14.454ptw\in W_{% v_{0}}(v).italic_G start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_v ) ∩ italic_B ( italic_w ) ≠ ∅ iff italic_w ∈ italic_W start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) .

Our motivation for Theorem 2.1 comes from Poisson geometry: when G0subscript𝐺0G_{0}italic_G start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT is connected, it is shown in [7] that there is a Poisson structure ΠΠ\Piroman_Π on {\mathcal{B}}caligraphic_B such that the connected components of intersections of G0subscript𝐺0G_{0}italic_G start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT-orbits and B𝐵Bitalic_B-orbits in {\mathcal{B}}caligraphic_B are precisely the H0subscript𝐻0H_{0}italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT-orbits of symplectic leaves of ΠΠ\Piroman_Π, where H0=BG0subscript𝐻0𝐵subscript𝐺0H_{0}=B\cap G_{0}italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_B ∩ italic_G start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT is a maximally compact Cartan subgroup of G0subscript𝐺0G_{0}italic_G start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT. Thus, one first needs to know when the intersection of a G0subscript𝐺0G_{0}italic_G start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT-orbit and a B𝐵Bitalic_B-orbit in {\mathcal{B}}caligraphic_B is non-empty. In view of (2.5), Theorem 2.1 gives a complete answer to this question in terms of the set Yv0(v)subscript𝑌subscript𝑣0𝑣Y_{v_{0}}(v)italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) and the Bruhat order on W𝑊Witalic_W. Further applications to the Poisson structure ΠΠ\Piroman_Π on {\mathcal{B}}caligraphic_B will appear elsewhere.

2.4. Yv0(v)subscript𝑌subscript𝑣0𝑣Y_{v_{0}}(v)italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) and Zv0(v)subscript𝑍subscript𝑣0𝑣Z_{v_{0}}(v)italic_Z start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) as invariants of K(v)𝐾𝑣K(v)italic_K ( italic_v )

Our second main result, the following Theorem 2.2, implies that the pair (ϕ(v),Yv0(v))italic-ϕ𝑣subscript𝑌subscript𝑣0𝑣(\phi(v),Y_{v_{0}}(v))( italic_ϕ ( italic_v ) , italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) ) forms a complete invariant for vV𝑣𝑉v\in Vitalic_v ∈ italic_V, where ϕ:VW:italic-ϕ𝑉𝑊\phi:V\to Witalic_ϕ : italic_V → italic_W is the Springer map (see §§\lx@sectionsign§6.6).

Theorem 2.2.

Let v0V0subscript𝑣0subscript𝑉0v_{0}\in V_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and v,vV𝑣superscript𝑣normal-′𝑉v,v^{\prime}\in Vitalic_v , italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_V. If ϕ(v)=ϕ(v)italic-ϕ𝑣italic-ϕsuperscript𝑣normal-′\phi(v)=\phi(v^{\prime})italic_ϕ ( italic_v ) = italic_ϕ ( italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) and Yv0(v)Yv0(v),subscript𝑌subscript𝑣0𝑣subscript𝑌subscript𝑣0superscript𝑣normal-′Y_{v_{0}}(v)\cap Y_{v_{0}}(v^{\prime})\neq\emptyset,italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) ∩ italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ≠ ∅ , then v=v𝑣superscript𝑣normal-′v=v^{\prime}italic_v = italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT.

Since Zv0(v)Yv0(v)subscript𝑍subscript𝑣0𝑣subscript𝑌subscript𝑣0𝑣Z_{v_{0}}(v)\subset Y_{v_{0}}(v)italic_Z start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) ⊂ italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) for any v0V0subscript𝑣0subscript𝑉0v_{0}\in V_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and vV𝑣𝑉v\in Vitalic_v ∈ italic_V, one also has

Corollary 2.3.

Let v0V0subscript𝑣0subscript𝑉0v_{0}\in V_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and v,vV𝑣superscript𝑣normal-′𝑉v,v^{\prime}\in Vitalic_v , italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_V. If ϕ(v)=ϕ(v)italic-ϕ𝑣italic-ϕsuperscript𝑣normal-′\phi(v)=\phi(v^{\prime})italic_ϕ ( italic_v ) = italic_ϕ ( italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) and Zv0(v)Zv0(v),subscript𝑍subscript𝑣0𝑣subscript𝑍subscript𝑣0superscript𝑣normal-′Z_{v_{0}}(v)\cap Z_{v_{0}}(v^{\prime})\neq\emptyset,italic_Z start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) ∩ italic_Z start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ≠ ∅ , then v=v𝑣superscript𝑣normal-′v=v^{\prime}italic_v = italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT.

2.5. Subexpressions and admissible paths

Reduced decompositions for elements in V𝑉Vitalic_V and their subexpressions are introduced in [12, §§\lx@sectionsign§5] and [13, §§\lx@sectionsign§3, §§\lx@sectionsign§4] (and see §§\lx@sectionsign§6.11 for details). In particular, if v,vV𝑣superscript𝑣𝑉v,v^{\prime}\in Vitalic_v , italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_V and if (𝐯,𝐬)superscript𝐯superscript𝐬({\bf v}^{\prime},{\bf s}^{\prime})( bold_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , bold_s start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) is any reduced decomposition of vsuperscript𝑣v^{\prime}italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT, then [13, Proposition 4.4] vv𝑣superscript𝑣v\leq v^{\prime}italic_v ≤ italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT if and only if there exists a subexpression of (𝐯,𝐬)superscript𝐯superscript𝐬({\bf v}^{\prime},{\bf s}^{\prime})( bold_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , bold_s start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) with final term v𝑣vitalic_v.

Let v0V0subscript𝑣0subscript𝑉0v_{0}\in V_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and vV𝑣𝑉v\in Vitalic_v ∈ italic_V. We show that if yYv0(v)𝑦subscript𝑌subscript𝑣0𝑣y\in Y_{v_{0}}(v)italic_y ∈ italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ), then every reduced decomposition of m(y)v0𝑚𝑦subscript𝑣0m(y)\!\cdot\!v_{0}italic_m ( italic_y ) ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT coming from a reduced word of y𝑦yitalic_y has exactly one subexpression with final term v𝑣vitalic_v. See Proposition 7.6 for details.

For v0V0subscript𝑣0subscript𝑉0v_{0}\in V_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and vV𝑣𝑉v\in Vitalic_v ∈ italic_V, we define an admissible path from v0subscript𝑣0v_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT to v𝑣vitalic_v to be a pair (𝐯,𝐬)𝐯𝐬({\bf v},{\bf s})( bold_v , bold_s ), where 𝐯=(v0,v1,,vk)𝐯subscript𝑣0subscript𝑣1subscript𝑣𝑘{\bf v}=(v_{0},v_{1},\ldots,v_{k})bold_v = ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_v start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) is a sequence in V𝑉Vitalic_V and 𝐬=(s1,,sk)𝐬subscript𝑠1subscript𝑠𝑘{\bf s}=(s_{1},\ldots,s_{k})bold_s = ( italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_s start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) is a sequence in S𝑆Sitalic_S, such that for each j[1,k]𝑗1𝑘j\in[1,k]italic_j ∈ [ 1 , italic_k ], vjsubscript𝑣𝑗v_{j}italic_v start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT is a certain type of either the cross action or the monoidal action of sjsubscript𝑠𝑗s_{j}italic_s start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT on vj-1subscript𝑣𝑗1v_{j-1}italic_v start_POSTSUBSCRIPT italic_j - 1 end_POSTSUBSCRIPT (see Definition 8.1). For such a path (𝐯,𝐬)𝐯𝐬({\bf v},{\bf s})( bold_v , bold_s ), let y(𝐯,𝐬)=sks1W𝑦𝐯𝐬subscript𝑠𝑘subscript𝑠1𝑊y({\bf v},{\bf s})=s_{k}\cdots s_{1}\in Witalic_y ( bold_v , bold_s ) = italic_s start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ⋯ italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∈ italic_W. We also define the set 𝒫min(v0,v)subscript𝒫minsubscript𝑣0𝑣{\mathcal{P}}_{\rm min}(v_{0},v)caligraphic_P start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v ) (resp. 𝒫short(v0,v)subscript𝒫shortsubscript𝑣0𝑣{\mathcal{P}}_{{\rm short}}(v_{0},v)caligraphic_P start_POSTSUBSCRIPT roman_short end_POSTSUBSCRIPT ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v )) of minimal (resp. shortest) paths from v0subscript𝑣0v_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT to v𝑣vitalic_v. We prove (see Corollary 8.8 and Corollary 8.10) that for any v0V0subscript𝑣0subscript𝑉0v_{0}\in V_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and vV𝑣𝑉v\in Vitalic_v ∈ italic_V,

Yv0(v)subscript𝑌subscript𝑣0𝑣\displaystyle Y_{v_{0}}(v)italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) ={y(𝐯,𝐬):(𝐯,𝐬)𝒫min(v0,v)}absentconditional-set𝑦𝐯𝐬𝐯𝐬subscript𝒫minsubscript𝑣0𝑣\displaystyle=\{y({\bf v},{\bf s}):({\bf v},{\bf s})\in{\mathcal{P}}_{\rm min}% (v_{0},v)\}= { italic_y ( bold_v , bold_s ) : ( bold_v , bold_s ) ∈ caligraphic_P start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v ) }
Zv0(v)subscript𝑍subscript𝑣0𝑣\displaystyle Z_{v_{0}}(v)italic_Z start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) ={y(𝐯,𝐬):(𝐯,𝐬)𝒫short(v0,v)}.absentconditional-set𝑦𝐯𝐬𝐯𝐬subscript𝒫shortsubscript𝑣0𝑣\displaystyle=\{y({\bf v},{\bf s}):({\bf v},{\bf s})\in{\mathcal{P}}_{\rm short% }(v_{0},v)\}.= { italic_y ( bold_v , bold_s ) : ( bold_v , bold_s ) ∈ caligraphic_P start_POSTSUBSCRIPT roman_short end_POSTSUBSCRIPT ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v ) } .

We believe that the minimal and shortest paths defined in this paper will have other applications to the study of K𝐾Kitalic_K-orbit closures in {\mathcal{B}}caligraphic_B.

3. Review on K𝐾Kitalic_K-orbits in {\mathcal{B}}caligraphic_B, I

Following [13, §§\lx@sectionsign§1.7], we review in this section the canonical Weyl group W𝑊Witalic_W of G𝐺Gitalic_G with its set S𝑆Sitalic_S of canonical generators and the action of the monoid M(W,S)𝑀𝑊𝑆M(W,S)italic_M ( italic_W , italic_S ) on V𝑉Vitalic_V. More structures on V𝑉Vitalic_V will be reviewed in §§\lx@sectionsign§6.

3.1. Notation

If Q𝑄Qitalic_Q is a subgroup of G𝐺Gitalic_G and gG𝑔𝐺g\in Gitalic_g ∈ italic_G, Qgsuperscript𝑄𝑔Q^{g}italic_Q start_POSTSUPERSCRIPT italic_g end_POSTSUPERSCRIPT denotes the subgroup g-1Qgsuperscript𝑔1𝑄𝑔g^{-1}Qgitalic_g start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_Q italic_g of G𝐺Gitalic_G. We will consider the right action of G𝐺Gitalic_G on the flag variety {\mathcal{B}}caligraphic_B by ×G:(B,g)Bg:𝐺maps-to𝐵𝑔superscript𝐵𝑔{\mathcal{B}}\times G\to{\mathcal{B}}:(B,g)\mapsto B^{g}caligraphic_B × italic_G → caligraphic_B : ( italic_B , italic_g ) ↦ italic_B start_POSTSUPERSCRIPT italic_g end_POSTSUPERSCRIPT for B𝐵B\in{\mathcal{B}}italic_B ∈ caligraphic_B and gG𝑔𝐺g\in Gitalic_g ∈ italic_G.

If a group L𝐿Litalic_L acts on a set X𝑋Xitalic_X from the left (resp. right), we will denote by L\X\𝐿𝑋L\backslash Xitalic_L \ italic_X (resp. X/L𝑋𝐿X/Litalic_X / italic_L) the set of L𝐿Litalic_L-orbits on X𝑋Xitalic_X.

Throughout the paper and unless otherwise specified, for a subset X𝑋Xitalic_X of G𝐺Gitalic_G (resp. {\mathcal{B}}caligraphic_B and ×{\mathcal{B}}\times{\mathcal{B}}caligraphic_B × caligraphic_B), X¯¯𝑋\overline{X}¯ start_ARG italic_X end_ARG denotes the Zariski closure of X𝑋Xitalic_X in G𝐺Gitalic_G (resp. {\mathcal{B}}caligraphic_B and ×{\mathcal{B}}\times{\mathcal{B}}caligraphic_B × caligraphic_B).

3.2. The variety 𝒞𝒞{\mathcal{C}}caligraphic_C

Let 𝒞𝒞{\mathcal{C}}caligraphic_C be the set of all pairs (B,H)𝐵𝐻(B,H)( italic_B , italic_H ), where B𝐵B\in{\mathcal{B}}italic_B ∈ caligraphic_B and HB𝐻𝐵H\subset Bitalic_H ⊂ italic_B is a maximal torus of G𝐺Gitalic_G. Then G𝐺Gitalic_G acts on 𝒞𝒞{\mathcal{C}}caligraphic_C transitively from the right by

(3.1) 𝒞×G𝒞:(B,H)g=(Bg,Hg),(B,H)𝒞,gG.:𝒞𝐺𝒞formulae-sequencesuperscript𝐵𝐻𝑔superscript𝐵𝑔superscript𝐻𝑔formulae-sequence𝐵𝐻𝒞𝑔𝐺{\mathcal{C}}\times G\longrightarrow{\mathcal{C}}:\;\;(B,H)^{g}=(B^{g},\;H^{g}% ),\hskip 14.454pt(B,H)\in{\mathcal{C}},\,g\in G.caligraphic_C × italic_G ⟶ caligraphic_C : ( italic_B , italic_H ) start_POSTSUPERSCRIPT italic_g end_POSTSUPERSCRIPT = ( italic_B start_POSTSUPERSCRIPT italic_g end_POSTSUPERSCRIPT , italic_H start_POSTSUPERSCRIPT italic_g end_POSTSUPERSCRIPT ) , ( italic_B , italic_H ) ∈ caligraphic_C , italic_g ∈ italic_G .

The stabilizer subgroup of G𝐺Gitalic_G at (B,H)𝒞𝐵𝐻𝒞(B,H)\in{\mathcal{C}}( italic_B , italic_H ) ∈ caligraphic_C is H𝐻Hitalic_H. Thus for each (B,H)𝒞𝐵𝐻𝒞(B,H)\in{\mathcal{C}}( italic_B , italic_H ) ∈ caligraphic_C, one has the G𝐺Gitalic_G-equivariant identification

(3.2) CB,H:H\G𝒞:Hg(Bg,Hg),gG.:subscript𝐶𝐵𝐻\𝐻𝐺𝒞:formulae-sequence𝐻𝑔superscript𝐵𝑔superscript𝐻𝑔𝑔𝐺C_{B,H}:\;\;\;H\backslash G\longrightarrow{\mathcal{C}}:\;\;\;Hg\longmapsto(B^% {g},H^{g}),\hskip 14.454ptg\in G.italic_C start_POSTSUBSCRIPT italic_B , italic_H end_POSTSUBSCRIPT : italic_H \ italic_G ⟶ caligraphic_C : italic_H italic_g ⟼ ( italic_B start_POSTSUPERSCRIPT italic_g end_POSTSUPERSCRIPT , italic_H start_POSTSUPERSCRIPT italic_g end_POSTSUPERSCRIPT ) , italic_g ∈ italic_G .

For (B,H)𝒞𝐵𝐻𝒞(B,H)\in{\mathcal{C}}( italic_B , italic_H ) ∈ caligraphic_C, let NG(H)subscript𝑁𝐺𝐻N_{G}(H)italic_N start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_H ) be the normalizer of H𝐻Hitalic_H in G𝐺Gitalic_G, let WH=NG(H)/Hsubscript𝑊𝐻subscript𝑁𝐺𝐻𝐻W_{H}=N_{G}(H)/Hitalic_W start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT = italic_N start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_H ) / italic_H be the Weyl group of (G,H)𝐺𝐻(G,H)( italic_G , italic_H ), and let SB,Hsubscript𝑆𝐵𝐻S_{B,H}italic_S start_POSTSUBSCRIPT italic_B , italic_H end_POSTSUBSCRIPT be the set of generators of WHsubscript𝑊𝐻W_{H}italic_W start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT defined by the simple roots of B𝐵Bitalic_B relative to H𝐻Hitalic_H.

Let (B,H),(B,H)𝒞𝐵𝐻superscript𝐵superscript𝐻𝒞(B,H),(B^{\prime},H^{\prime})\in{\mathcal{C}}( italic_B , italic_H ) , ( italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_H start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ∈ caligraphic_C. Let gG𝑔𝐺g\in Gitalic_g ∈ italic_G be such that B=Bgsuperscript𝐵superscript𝐵𝑔B^{\prime}=B^{g}italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_B start_POSTSUPERSCRIPT italic_g end_POSTSUPERSCRIPT and H=Hgsuperscript𝐻superscript𝐻𝑔H^{\prime}=H^{g}italic_H start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_H start_POSTSUPERSCRIPT italic_g end_POSTSUPERSCRIPT. Since g𝑔gitalic_g is unique up to the left multiplication by an element in H𝐻Hitalic_H, one has a well-defined isomorphism of tori

(3.3) TB,HB,H:HH:hg-1hg,hH.:superscriptsubscript𝑇𝐵𝐻superscript𝐵superscript𝐻𝐻superscript𝐻:formulae-sequencesuperscript𝑔1𝑔𝐻T_{B,H}^{B^{\prime},H^{\prime}}:\;\;H\longrightarrow H^{\prime}:\;\;h% \longmapsto g^{-1}hg,\hskip 14.454pth\in H.italic_T start_POSTSUBSCRIPT italic_B , italic_H end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_H start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT : italic_H ⟶ italic_H start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT : italic_h ⟼ italic_g start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_h italic_g , italic_h ∈ italic_H .

Moreover, although the map NG(H)NG(H):ng-1ng:subscript𝑁𝐺𝐻subscript𝑁𝐺superscript𝐻𝑛superscript𝑔1𝑛𝑔N_{G}(H)\to N_{G}(H^{\prime}):n\to g^{-1}ngitalic_N start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_H ) → italic_N start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_H start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) : italic_n → italic_g start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_n italic_g depends on the choice of g𝑔gitalic_g, the group isomorphism

ηB,HB,H:WHWH:nHg-1ngH,nNG(H):superscriptsubscript𝜂𝐵𝐻superscript𝐵superscript𝐻subscript𝑊𝐻subscript𝑊superscript𝐻:formulae-sequence𝑛𝐻superscript𝑔1𝑛𝑔superscript𝐻𝑛subscript𝑁𝐺𝐻\eta_{B,H}^{B^{\prime},H^{\prime}}:\;\;W_{H}\longrightarrow W_{H^{\prime}}:\;% \;nH\longmapsto g^{-1}ngH^{\prime},\hskip 14.454ptn\in N_{G}(H)italic_η start_POSTSUBSCRIPT italic_B , italic_H end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_H start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT : italic_W start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT ⟶ italic_W start_POSTSUBSCRIPT italic_H start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT : italic_n italic_H ⟼ italic_g start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_n italic_g italic_H start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_n ∈ italic_N start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_H )

does not and is thus well-defined. Since ηB,HB,H(SB,H)=SB,Hsuperscriptsubscript𝜂𝐵𝐻superscript𝐵superscript𝐻subscript𝑆𝐵𝐻subscript𝑆superscript𝐵superscript𝐻\eta_{B,H}^{B^{\prime},H^{\prime}}(S_{B,H})=S_{B^{\prime},H^{\prime}}italic_η start_POSTSUBSCRIPT italic_B , italic_H end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_H start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT ( italic_S start_POSTSUBSCRIPT italic_B , italic_H end_POSTSUBSCRIPT ) = italic_S start_POSTSUBSCRIPT italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_H start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT, one has the isomorphism ηB,HB,H:(WH,SB,H)(WH,SB,H):superscriptsubscript𝜂𝐵𝐻superscript𝐵superscript𝐻subscript𝑊𝐻subscript𝑆𝐵𝐻subscript𝑊superscript𝐻subscript𝑆superscript𝐵superscript𝐻\eta_{B,H}^{B^{\prime},H^{\prime}}:(W_{H},S_{B,H})\to(W_{H^{\prime}},S_{B^{% \prime},H^{\prime}})italic_η start_POSTSUBSCRIPT italic_B , italic_H end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_H start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT : ( italic_W start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT , italic_S start_POSTSUBSCRIPT italic_B , italic_H end_POSTSUBSCRIPT ) → ( italic_W start_POSTSUBSCRIPT italic_H start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT , italic_S start_POSTSUBSCRIPT italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_H start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) of Coxeter groups.

3.3. The canonical Weyl group

Let W=(×)/G𝑊𝐺W=({\mathcal{B}}\times{\mathcal{B}})/Gitalic_W = ( caligraphic_B × caligraphic_B ) / italic_G be the set of G𝐺Gitalic_G-orbits on ×{\mathcal{B}}\times{\mathcal{B}}caligraphic_B × caligraphic_B for the diagonal G𝐺Gitalic_G-action, and let p:×W:𝑝𝑊p:{\mathcal{B}}\times{\mathcal{B}}\to Witalic_p : caligraphic_B × caligraphic_B → italic_W be the natural projection. Let (B,H)𝒞𝐵𝐻𝒞(B,H)\in{\mathcal{C}}( italic_B , italic_H ) ∈ caligraphic_C. Then the map

(3.4) ηB,H:WHW:nHp(Bn,B),nNG(H):subscript𝜂𝐵𝐻subscript𝑊𝐻𝑊:formulae-sequence𝑛𝐻𝑝superscript𝐵𝑛𝐵𝑛subscript𝑁𝐺𝐻\eta_{B,H}:\;\;\;W_{H}\longrightarrow W:\;\;nH\longmapsto p(B^{n},B),\hskip 14% .454ptn\in N_{G}(H)italic_η start_POSTSUBSCRIPT italic_B , italic_H end_POSTSUBSCRIPT : italic_W start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT ⟶ italic_W : italic_n italic_H ⟼ italic_p ( italic_B start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT , italic_B ) , italic_n ∈ italic_N start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_H )

is bijective. It is straightforward to check that for another (B,H)𝒞superscript𝐵superscript𝐻𝒞(B^{\prime},H^{\prime})\in{\mathcal{C}}( italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_H start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ∈ caligraphic_C,

ηB,H-1ηB,H=ηB,HB,H:WHWH.:superscriptsubscript𝜂superscript𝐵superscript𝐻1subscript𝜂𝐵𝐻superscriptsubscript𝜂𝐵𝐻superscript𝐵superscript𝐻subscript𝑊𝐻subscript𝑊superscript𝐻\eta_{B^{\prime},H^{\prime}}^{-1}\circ\eta_{B,H}=\eta_{B,H}^{B^{\prime},H^{% \prime}}:\;\;\;W_{H}\longrightarrow W_{H^{\prime}}.italic_η start_POSTSUBSCRIPT italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_H start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ∘ italic_η start_POSTSUBSCRIPT italic_B , italic_H end_POSTSUBSCRIPT = italic_η start_POSTSUBSCRIPT italic_B , italic_H end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_H start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT : italic_W start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT ⟶ italic_W start_POSTSUBSCRIPT italic_H start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT .

Thus there is a well-defined group structure on W𝑊Witalic_W such that ηB,H:WHW:subscript𝜂𝐵𝐻subscript𝑊𝐻𝑊\eta_{B,H}:W_{H}\to Witalic_η start_POSTSUBSCRIPT italic_B , italic_H end_POSTSUBSCRIPT : italic_W start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT → italic_W is a group isomorphism for every (B,H)𝒞𝐵𝐻𝒞(B,H)\in{\mathcal{C}}( italic_B , italic_H ) ∈ caligraphic_C. Let S=ηB,H(SB,H)W𝑆subscript𝜂𝐵𝐻subscript𝑆𝐵𝐻𝑊S=\eta_{B,H}(S_{B,H})\subset Witalic_S = italic_η start_POSTSUBSCRIPT italic_B , italic_H end_POSTSUBSCRIPT ( italic_S start_POSTSUBSCRIPT italic_B , italic_H end_POSTSUBSCRIPT ) ⊂ italic_W for any (B,H)𝒞𝐵𝐻𝒞(B,H)\in{\mathcal{C}}( italic_B , italic_H ) ∈ caligraphic_C. Then S𝑆Sitalic_S is a set of generators for W𝑊Witalic_W and is independent of the choice of (B,H)𝒞𝐵𝐻𝒞(B,H)\in{\mathcal{C}}( italic_B , italic_H ) ∈ caligraphic_C. The Coxeter group (W,S)𝑊𝑆(W,S)( italic_W , italic_S ) is called the canonical Weyl group of G𝐺Gitalic_G.

For wW𝑤𝑊w\in Witalic_w ∈ italic_W, a reduced word of w𝑤witalic_w is a shortest expression w=s1sl(w)𝑤subscript𝑠1subscript𝑠𝑙𝑤w=s_{1}\cdots s_{l(w)}italic_w = italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⋯ italic_s start_POSTSUBSCRIPT italic_l ( italic_w ) end_POSTSUBSCRIPT of w𝑤witalic_w as a product of elements in S𝑆Sitalic_S, and l(w)𝑙𝑤l(w)italic_l ( italic_w ) is called the length of w𝑤witalic_w.

3.4. The Bruhat order on W𝑊Witalic_W

For wW𝑤𝑊w\in Witalic_w ∈ italic_W, let

𝒪(w)={(B,B)×:p(B,B)=w}×𝒪𝑤conditional-setsuperscript𝐵𝐵𝑝superscript𝐵𝐵𝑤{\mathcal{O}}(w)=\{(B^{\prime},B)\in{\mathcal{B}}\times{\mathcal{B}}:\,p(B^{% \prime},B)=w\}\subset{\mathcal{B}}\times{\mathcal{B}}caligraphic_O ( italic_w ) = { ( italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_B ) ∈ caligraphic_B × caligraphic_B : italic_p ( italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_B ) = italic_w } ⊂ caligraphic_B × caligraphic_B

be the corresponding G𝐺Gitalic_G-orbit in ×{\mathcal{B}}\times{\mathcal{B}}caligraphic_B × caligraphic_B. The Bruhat order on W𝑊Witalic_W is defined by

ww  if  𝒪(w)𝒪(w)¯,w,wW.formulae-sequence𝑤superscript𝑤ifformulae-sequence𝒪𝑤¯𝒪superscript𝑤𝑤superscript𝑤𝑊w\leq w^{\prime}\hskip 14.454pt\mbox{if}\hskip 14.454pt{\mathcal{O}}(w)\subset% \overline{{\mathcal{O}}(w^{\prime})},\;\;\;w,w^{\prime}\in W.italic_w ≤ italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT if caligraphic_O ( italic_w ) ⊂ ¯ start_ARG caligraphic_O ( italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) end_ARG , italic_w , italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_W .

If ww𝑤superscript𝑤w\leq w^{\prime}italic_w ≤ italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT and ww𝑤superscript𝑤w\neq w^{\prime}italic_w ≠ italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT, we will write w<w𝑤superscript𝑤w<w^{\prime}italic_w < italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT or w>wsuperscript𝑤𝑤w^{\prime}>witalic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT > italic_w. It is well-known [6, Théorème 3.13] that for w,wW𝑤superscript𝑤𝑊w,w^{\prime}\in Witalic_w , italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_W, wwsuperscript𝑤𝑤w^{\prime}\leq witalic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≤ italic_w if and only if there is a reduced word w=s1sl(w)𝑤subscript𝑠1subscript𝑠𝑙𝑤w=s_{1}\cdots s_{l(w)}italic_w = italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⋯ italic_s start_POSTSUBSCRIPT italic_l ( italic_w ) end_POSTSUBSCRIPT of w𝑤witalic_w such that w=si1sipsuperscript𝑤subscript𝑠subscript𝑖1subscript𝑠subscript𝑖𝑝w^{\prime}=s_{i_{1}}\cdots s_{i_{p}}italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_s start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⋯ italic_s start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT end_POSTSUBSCRIPT for some 1i1<<ipl(w)1subscript𝑖1subscript𝑖𝑝𝑙𝑤1\leq i_{1}<\cdots<i_{p}\leq l(w)1 ≤ italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT < ⋯ < italic_i start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ≤ italic_l ( italic_w ).

For a subset W1subscript𝑊1W_{1}italic_W start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT of W𝑊Witalic_W, recall from §§\lx@sectionsign§2.2 that min(W1)subscript𝑊1\min(W_{1})roman_min ( italic_W start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) is the set of minimal elements in W1subscript𝑊1W_{1}italic_W start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT with respect to the Bruhat order and that minl(W1)subscript𝑙subscript𝑊1\min_{l}(W_{1})roman_min start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ( italic_W start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) is the set of minimal length elements in W1subscript𝑊1W_{1}italic_W start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT. The following Lemma 3.1 will be used in the proof of Proposition 4.5.

Lemma 3.1.

Let W1subscript𝑊1W_{1}italic_W start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and W2subscript𝑊2W_{2}italic_W start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT be two subsets of W𝑊Witalic_W such that min(W2)W1W2subscript𝑊2subscript𝑊1subscript𝑊2\min(W_{2})\subset W_{1}\subset W_{2}roman_min ( italic_W start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ⊂ italic_W start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⊂ italic_W start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT. Then min(W1)=min(W2)subscript𝑊1subscript𝑊2\min(W_{1})=\min(W_{2})roman_min ( italic_W start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) = roman_min ( italic_W start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) and minl(W1)=minl(W2)subscript𝑙subscript𝑊1subscript𝑙subscript𝑊2\min_{l}(W_{1})=\min_{l}(W_{2})roman_min start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ( italic_W start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) = roman_min start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ( italic_W start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ).

3.5. Orbits in {\mathcal{B}}caligraphic_B under a Borel subgroup

Let B𝐵B\in{\mathcal{B}}italic_B ∈ caligraphic_B. The set of B𝐵Bitalic_B-orbits in {\mathcal{B}}caligraphic_B is naturally indexed by W𝑊Witalic_W. Indeed, for wW𝑤𝑊w\in Witalic_w ∈ italic_W, let

(3.5) B(w)={B:p(B,B)=w}.𝐵𝑤conditional-setsuperscript𝐵𝑝superscript𝐵𝐵𝑤B(w)=\{B^{\prime}\in{\mathcal{B}}:\,p(B^{\prime},B)=w\}\subset{\mathcal{B}}.italic_B ( italic_w ) = { italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ caligraphic_B : italic_p ( italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_B ) = italic_w } ⊂ caligraphic_B .

Then B(w)𝐵𝑤B(w)italic_B ( italic_w ) is a single B𝐵Bitalic_B-orbit in {\mathcal{B}}caligraphic_B, and the map wB(w)maps-to𝑤𝐵𝑤w\mapsto B(w)italic_w ↦ italic_B ( italic_w ) is a bijection from W𝑊Witalic_W to the set of all B𝐵Bitalic_B-orbits in {\mathcal{B}}caligraphic_B.

Define qB:G:subscript𝑞𝐵𝐺q_{B}:G\to{\mathcal{B}}italic_q start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT : italic_G → caligraphic_B by qB(g)=Bgsubscript𝑞𝐵𝑔superscript𝐵𝑔q_{B}(g)=B^{g}italic_q start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_g ) = italic_B start_POSTSUPERSCRIPT italic_g end_POSTSUPERSCRIPT for gG𝑔𝐺g\in Gitalic_g ∈ italic_G. For wW𝑤𝑊w\in Witalic_w ∈ italic_W, let

(3.6) BwB=qB-1(B(w))={gG:p(Bg,B)=w}G.𝐵𝑤𝐵superscriptsubscript𝑞𝐵1𝐵𝑤conditional-set𝑔𝐺𝑝superscript𝐵𝑔𝐵𝑤𝐺BwB=q_{B}^{-1}(B(w))=\{g\in G:p(B^{g},B)=w\}\subset G.italic_B italic_w italic_B = italic_q start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_B ( italic_w ) ) = { italic_g ∈ italic_G : italic_p ( italic_B start_POSTSUPERSCRIPT italic_g end_POSTSUPERSCRIPT , italic_B ) = italic_w } ⊂ italic_G .

Then BwB𝐵𝑤𝐵BwBitalic_B italic_w italic_B be the single (B,B)𝐵𝐵(B,B)( italic_B , italic_B )-double coset in G𝐺Gitalic_G. Moreover, for w,wW𝑤superscript𝑤𝑊w,w^{\prime}\in Witalic_w , italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_W,

ww  iff  B(w)B(w)¯  iff  BwBBwB¯.formulae-sequence𝑤superscript𝑤iffformulae-sequence𝐵𝑤¯𝐵superscript𝑤iff𝐵𝑤𝐵¯𝐵superscript𝑤𝐵w\leq w^{\prime}\hskip 14.454pt\mbox{iff}\hskip 14.454ptB(w)\subset\overline{B% (w^{\prime})}\hskip 14.454pt\mbox{iff}\hskip 14.454ptBwB\subset\overline{Bw^{% \prime}B}.italic_w ≤ italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT iff italic_B ( italic_w ) ⊂ ¯ start_ARG italic_B ( italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) end_ARG iff italic_B italic_w italic_B ⊂ ¯ start_ARG italic_B italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_B end_ARG .

3.6. The monoidal action of M(W,S)𝑀𝑊𝑆M(W,S)italic_M ( italic_W , italic_S ) on V𝑉Vitalic_V

Our references for this subsection are [12, §§\lx@sectionsign§4] and [13, §§\lx@sectionsign§3]. The monoid M(W,S)𝑀𝑊𝑆M(W,S)italic_M ( italic_W , italic_S ) associated to the Coxeter group (W,S)𝑊𝑆(W,S)( italic_W , italic_S ) is M(W,S)={m(w):wW}𝑀𝑊𝑆conditional-set𝑚𝑤𝑤𝑊M(W,S)=\{m(w):w\in W\}italic_M ( italic_W , italic_S ) = { italic_m ( italic_w ) : italic_w ∈ italic_W } as a set, with the monoidal product given by

(3.7) m(s)m(w)={m(sw),ifsw>wm(w),ifsw<w,sS,wW.formulae-sequence𝑚𝑠𝑚𝑤cases𝑚𝑠𝑤if𝑠𝑤𝑤𝑚𝑤if𝑠𝑤𝑤formulae-sequence𝑠𝑆𝑤𝑊m(s)m(w)=\begin{cases}m(sw),&\;\;\;\mbox{if}\;\;sw>w\\ m(w),&\;\;\;\mbox{if}\;\;sw<w\end{cases},\hskip 14.454pts\in S,\;w\in W.italic_m ( italic_s ) italic_m ( italic_w ) = { start_ROW start_CELL italic_m ( italic_s italic_w ) , end_CELL start_CELL if italic_s italic_w > italic_w end_CELL end_ROW start_ROW start_CELL italic_m ( italic_w ) , end_CELL start_CELL if italic_s italic_w < italic_w end_CELL end_ROW , italic_s ∈ italic_S , italic_w ∈ italic_W .

Alternatively, let B𝐵B\in{\mathcal{B}}italic_B ∈ caligraphic_B, and let BwB=qB-1(B(w))G𝐵𝑤𝐵superscriptsubscript𝑞𝐵1𝐵𝑤𝐺BwB=q_{B}^{-1}(B(w))\subset Gitalic_B italic_w italic_B = italic_q start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_B ( italic_w ) ) ⊂ italic_G for wW𝑤𝑊w\in Witalic_w ∈ italic_W as in (3.6). Define :W×WWfragments:WWW\ast:W\times W\to W∗ : italic_W × italic_W → italic_W such that

(3.8) B(w1w2)B¯=(Bw1B)(Bw2B)¯,w1,w2W.formulae-sequence¯𝐵subscript𝑤1subscript𝑤2𝐵¯𝐵subscript𝑤1𝐵𝐵subscript𝑤2𝐵subscript𝑤1subscript𝑤2𝑊\overline{B(w_{1}\ast w_{2})B}=\overline{(Bw_{1}B)(Bw_{2}B)},\hskip 14.454ptw_% {1},w_{2}\in W.¯ start_ARG italic_B ( italic_w start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∗ italic_w start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) italic_B end_ARG = ¯ start_ARG ( italic_B italic_w start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_B ) ( italic_B italic_w start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_B ) end_ARG , italic_w start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∈ italic_W .

Then \ast is a monoidal product on W𝑊Witalic_W, independent of the choice of B𝐵B\in{\mathcal{B}}italic_B ∈ caligraphic_B, and (W,)M(W,S):wm(w):𝑊𝑀𝑊𝑆maps-to𝑤𝑚𝑤(W,\ast)\to M(W,S):w\mapsto m(w)( italic_W , ∗ ) → italic_M ( italic_W , italic_S ) : italic_w ↦ italic_m ( italic_w ) is an isomorphism of monoids [6, Proposition 3.18].

Let qK:GG/K:subscript𝑞𝐾𝐺𝐺𝐾q_{K}:G\to G/Kitalic_q start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT : italic_G → italic_G / italic_K be the natural projection. Let B𝐵B\in{\mathcal{B}}italic_B ∈ caligraphic_B, and for vV𝑣𝑉v\in Vitalic_v ∈ italic_V, let

BvK=qB-1(K(v))G.𝐵𝑣𝐾superscriptsubscript𝑞𝐵1𝐾𝑣𝐺BvK=q_{B}^{-1}(K(v))\subset G.italic_B italic_v italic_K = italic_q start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_K ( italic_v ) ) ⊂ italic_G .

Then qK(BvK)subscript𝑞𝐾𝐵𝑣𝐾q_{K}(BvK)italic_q start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT ( italic_B italic_v italic_K ), being a B𝐵Bitalic_B-orbit in G/K𝐺𝐾G/Kitalic_G / italic_K, is irreducible for each vV𝑣𝑉v\in Vitalic_v ∈ italic_V. For wW𝑤𝑊w\in Witalic_w ∈ italic_W and vV𝑣𝑉v\in Vitalic_v ∈ italic_V, since qK((BwB)(BvK))G/Ksubscript𝑞𝐾𝐵𝑤𝐵𝐵𝑣𝐾𝐺𝐾q_{K}((BwB)(BvK))\subset G/Kitalic_q start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT ( ( italic_B italic_w italic_B ) ( italic_B italic_v italic_K ) ) ⊂ italic_G / italic_K is irreducible and is a finite union of B𝐵Bitalic_B-orbits in G/K𝐺𝐾G/Kitalic_G / italic_K, there is a unique element in V𝑉Vitalic_V, denoted by m(w)v𝑚𝑤𝑣m(w)\!\cdot\!vitalic_m ( italic_w ) ⋅ italic_v, such that qK((BwB)(BvK)¯=qK(B(m(w)v)K)¯¯fragmentssubscript𝑞𝐾fragments(fragments(BwB)fragments(BvK)¯subscript𝑞𝐾𝐵𝑚𝑤𝑣𝐾\overline{q_{K}((BwB)(BvK)}=\overline{q_{K}(B(m(w)\!\cdot\!v)K)}¯ start_ARG italic_q start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT ( ( italic_B italic_w italic_B ) ( italic_B italic_v italic_K ) end_ARG = ¯ start_ARG italic_q start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT ( italic_B ( italic_m ( italic_w ) ⋅ italic_v ) italic_K ) end_ARG, where ¯¯absent\,\bar{}\,¯ start_ARG end_ARG denotes the Zariski closure in G/K𝐺𝐾G/Kitalic_G / italic_K. Consequently, one has, for any wW𝑤𝑊w\in Witalic_w ∈ italic_W and vV𝑣𝑉v\in Vitalic_v ∈ italic_V,

(3.9) B(m(w)v)K¯=(BwB)(BvK)¯.¯𝐵𝑚𝑤𝑣𝐾¯𝐵𝑤𝐵𝐵𝑣𝐾\overline{B(m(w)\!\cdot\!v)K}=\overline{(BwB)(BvK)}.¯ start_ARG italic_B ( italic_m ( italic_w ) ⋅ italic_v ) italic_K end_ARG = ¯ start_ARG ( italic_B italic_w italic_B ) ( italic_B italic_v italic_K ) end_ARG .
Lemma 3.2.

[12, §§\lx@sectionsign§4] The map

(3.10) M(W,S)×VV:(m(w),v)m(w)v,wW,vV,:𝑀𝑊𝑆𝑉𝑉formulae-sequence𝑚𝑤𝑣𝑚𝑤𝑣formulae-sequence𝑤𝑊𝑣𝑉M(W,S)\times V\longrightarrow V:\;\;\;(m(w),v)\longmapsto m(w)\!\cdot\!v,% \hskip 14.454ptw\in W,v\in V,italic_M ( italic_W , italic_S ) × italic_V ⟶ italic_V : ( italic_m ( italic_w ) , italic_v ) ⟼ italic_m ( italic_w ) ⋅ italic_v , italic_w ∈ italic_W , italic_v ∈ italic_V ,

is independent of the choice of B𝐵B\in{\mathcal{B}}italic_B ∈ caligraphic_B and defines a monoidal action of M(W,S)𝑀𝑊𝑆M(W,S)italic_M ( italic_W , italic_S ) on V𝑉Vitalic_V.

Proof.

Let B,g0Gformulae-sequence𝐵subscript𝑔0𝐺B\in{\mathcal{B}},g_{0}\in Gitalic_B ∈ caligraphic_B , italic_g start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ italic_G, and B=Bg0superscript𝐵superscript𝐵subscript𝑔0B^{\prime}=B^{g_{0}}italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_B start_POSTSUPERSCRIPT italic_g start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT. It follows from definitions that for any wW𝑤𝑊w\in Witalic_w ∈ italic_W and vV𝑣𝑉v\in Vitalic_v ∈ italic_V, BwB=g0-1BwBg0superscript𝐵𝑤superscript𝐵superscriptsubscript𝑔01𝐵𝑤𝐵subscript𝑔0B^{\prime}wB^{\prime}=g_{0}^{-1}BwBg_{0}italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_w italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_g start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_B italic_w italic_B italic_g start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and BvK=g0-1BvKsuperscript𝐵𝑣𝐾superscriptsubscript𝑔01𝐵𝑣𝐾B^{\prime}vK=g_{0}^{-1}BvKitalic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_v italic_K = italic_g start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_B italic_v italic_K. Thus, the map in (3.10) is independent of the choice of B𝐵Bitalic_B. Choosing any B𝐵B\in{\mathcal{B}}italic_B ∈ caligraphic_B and using the isomorphism (W,)M(W,S)𝑊𝑀𝑊𝑆(W,\ast)\to M(W,S)( italic_W , ∗ ) → italic_M ( italic_W , italic_S ) of monoids, one sees from (3.8) and (3.9) that the map in (3.10) defines an action of M(W,S)𝑀𝑊𝑆M(W,S)italic_M ( italic_W , italic_S ) on V𝑉Vitalic_V.

Q.E.D.

Lemma 3.3.

For any B,vVformulae-sequence𝐵𝑣𝑉B\in{\mathcal{B}},v\in Vitalic_B ∈ caligraphic_B , italic_v ∈ italic_V, and wW𝑤𝑊w\in Witalic_w ∈ italic_W, one has

B(m(w)v)K¯=BwB¯BvK¯.¯𝐵𝑚𝑤𝑣𝐾¯𝐵𝑤𝐵¯𝐵𝑣𝐾\overline{B(m(w)\!\cdot\!v)K}=\overline{BwB}\;\;\overline{BvK}.¯ start_ARG italic_B ( italic_m ( italic_w ) ⋅ italic_v ) italic_K end_ARG = ¯ start_ARG italic_B italic_w italic_B end_ARG ¯ start_ARG italic_B italic_v italic_K end_ARG .
Proof.

Let w=s1s2sl𝑤subscript𝑠1subscript𝑠2subscript𝑠𝑙w=s_{1}s_{2}\cdots s_{l}italic_w = italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_s start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⋯ italic_s start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT be a reduced word, and for each 1jl1𝑗𝑙1\leq j\leq l1 ≤ italic_j ≤ italic_l, let Pj=BBsjBsubscript𝑃𝑗𝐵𝐵subscript𝑠𝑗𝐵P_{j}=B\cup Bs_{j}Bitalic_P start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = italic_B ∪ italic_B italic_s start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_B so that Pjsubscript𝑃𝑗P_{j}italic_P start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT is a parabolic subgroup of G𝐺Gitalic_G. Then BwB¯=P1P2Pj¯𝐵𝑤𝐵subscript𝑃1subscript𝑃2subscript𝑃𝑗\overline{BwB}=P_{1}P_{2}\cdots P_{j}¯ start_ARG italic_B italic_w italic_B end_ARG = italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⋯ italic_P start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT (see, for example [6, Théorème 3.13]). By repeatedly applying [6, Lemma 2.12], one sees that BwB¯BvK¯¯𝐵𝑤𝐵¯𝐵𝑣𝐾\overline{BwB}\;\;\overline{BvK}¯ start_ARG italic_B italic_w italic_B end_ARG ¯ start_ARG italic_B italic_v italic_K end_ARG is closed in G𝐺Gitalic_G. Thus BwB¯BvK¯=(BwB)(BvK)¯=B(m(w)v)K¯¯𝐵𝑤𝐵¯𝐵𝑣𝐾¯𝐵𝑤𝐵𝐵𝑣𝐾¯𝐵𝑚𝑤𝑣𝐾\overline{BwB}\;\;\overline{BvK}=\overline{(BwB)(BvK)}=\overline{B(m(w)\!\cdot% \!v)K}¯ start_ARG italic_B italic_w italic_B end_ARG ¯ start_ARG italic_B italic_v italic_K end_ARG = ¯ start_ARG ( italic_B italic_w italic_B ) ( italic_B italic_v italic_K ) end_ARG = ¯ start_ARG italic_B ( italic_m ( italic_w ) ⋅ italic_v ) italic_K end_ARG.

Q.E.D.

Recall that V0={v0V:K(v0)is closed}subscript𝑉0conditional-setsubscript𝑣0𝑉𝐾subscript𝑣0is closedV_{0}=\{v_{0}\in V:K(v_{0})\subset{\mathcal{B}}\;\mbox{is closed}\}italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = { italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ italic_V : italic_K ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) ⊂ caligraphic_B is closed }. For a subset X𝑋Xitalic_X of {\mathcal{B}}caligraphic_B, let

XK={Bk:BX,kK}.𝑋𝐾conditional-setsuperscript𝐵𝑘formulae-sequence𝐵𝑋𝑘𝐾X\cdot K=\{B^{k}:\;B\in X,k\in K\}.italic_X ⋅ italic_K = { italic_B start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT : italic_B ∈ italic_X , italic_k ∈ italic_K } .
Lemma 3.4.

Let v0V0subscript𝑣0subscript𝑉0v_{0}\in V_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and let BK(v0)𝐵𝐾subscript𝑣0B\in K(v_{0})italic_B ∈ italic_K ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ). Then for any wW𝑤𝑊w\in Witalic_w ∈ italic_W,

K(m(w)v0)¯=B(w)K¯=B(w)¯K.¯𝐾𝑚𝑤subscript𝑣0¯𝐵𝑤𝐾¯𝐵𝑤𝐾\overline{K(m(w)\!\cdot\!v_{0})}=\overline{B(w)\cdot K}=\overline{B(w)}\cdot K.¯ start_ARG italic_K ( italic_m ( italic_w ) ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) end_ARG = ¯ start_ARG italic_B ( italic_w ) ⋅ italic_K end_ARG = ¯ start_ARG italic_B ( italic_w ) end_ARG ⋅ italic_K .
Proof.

Since K(v0)𝐾subscript𝑣0K(v_{0})italic_K ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) is closed in {\mathcal{B}}caligraphic_B, the double coset Bv0K=BK=qB-1(K(v0))𝐵subscript𝑣0𝐾𝐵𝐾superscriptsubscript𝑞𝐵1𝐾subscript𝑣0Bv_{0}K=BK=q_{B}^{-1}(K(v_{0}))italic_B italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_K = italic_B italic_K = italic_q start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_K ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) ) is closed in G𝐺Gitalic_G. By Lemma 3.3, one has

B(m(w)v0)K¯=BwB¯BK¯=BwB¯K.¯𝐵𝑚𝑤subscript𝑣0𝐾¯𝐵𝑤𝐵¯𝐵𝐾¯𝐵𝑤𝐵𝐾\overline{B(m(w)\!\cdot\!v_{0})K}=\overline{BwB}\;\overline{BK}=\overline{BwB}% \;K.¯ start_ARG italic_B ( italic_m ( italic_w ) ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) italic_K end_ARG = ¯ start_ARG italic_B italic_w italic_B end_ARG ¯ start_ARG italic_B italic_K end_ARG = ¯ start_ARG italic_B italic_w italic_B end_ARG italic_K .

Applying qB:G:subscript𝑞𝐵𝐺q_{B}:G\to{\mathcal{B}}italic_q start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT : italic_G → caligraphic_B, one proves Lemma 3.4.

Q.E.D.

Lemma 3.5.

If w,wW𝑤superscript𝑤normal-′𝑊w,w^{\prime}\in Witalic_w , italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_W and v,vV𝑣superscript𝑣normal-′𝑉v,v^{\prime}\in Vitalic_v , italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_V are such that ww𝑤superscript𝑤normal-′w\leq w^{\prime}italic_w ≤ italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT, vv𝑣superscript𝑣normal-′v\leq v^{\prime}italic_v ≤ italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT, then

m(w)vm(w)v  𝑎𝑛𝑑  m(w)vm(w)v.formulae-sequence𝑚𝑤𝑣𝑚superscript𝑤𝑣𝑎𝑛𝑑𝑚𝑤𝑣𝑚𝑤superscript𝑣m(w)\!\cdot\!v\leq m(w^{\prime})\!\cdot\!v\hskip 14.454pt\mbox{and}\hskip 14.4% 54ptm(w)\!\cdot\!v\leq m(w)\!\cdot\!v^{\prime}.italic_m ( italic_w ) ⋅ italic_v ≤ italic_m ( italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ⋅ italic_v and italic_m ( italic_w ) ⋅ italic_v ≤ italic_m ( italic_w ) ⋅ italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT .
Proof.

This is immediate from Lemma 3.3.

Q.E.D.

4. The geometrical and combinatorial definitions of Wv0(v)subscript𝑊subscript𝑣0𝑣W_{v_{0}}(v)italic_W start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v )

4.1. The two definitions of Wv0(v)subscript𝑊subscript𝑣0𝑣W_{v_{0}}(v)italic_W start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v )

Recall from §§\lx@sectionsign§2.1 that for v0V0subscript𝑣0subscript𝑉0v_{0}\in V_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and vV𝑣𝑉v\in Vitalic_v ∈ italic_V, the subset Wv0(v)subscript𝑊subscript𝑣0𝑣W_{v_{0}}(v)italic_W start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) of W𝑊Witalic_W is defined by

Wv0(v)={wW:K(v)B(w)¯},subscript𝑊subscript𝑣0𝑣conditional-set𝑤𝑊𝐾𝑣¯𝐵𝑤W_{v_{0}}(v)=\{w\in W:K(v)\cap\overline{B(w)}\neq\emptyset\},italic_W start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) = { italic_w ∈ italic_W : italic_K ( italic_v ) ∩ ¯ start_ARG italic_B ( italic_w ) end_ARG ≠ ∅ } ,

where B𝐵Bitalic_B is any Borel subgroup of G𝐺Gitalic_G contained in K(v0)𝐾subscript𝑣0K(v_{0})italic_K ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ).

Lemma 4.1.

For any v0Vsubscript𝑣0𝑉v_{0}\in Vitalic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ italic_V and vV𝑣𝑉v\in Vitalic_v ∈ italic_V, the set Wv0(v)subscript𝑊subscript𝑣0𝑣W_{v_{0}}(v)italic_W start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) is independent of the choice of BK(v0)𝐵𝐾subscript𝑣0B\in K(v_{0})italic_B ∈ italic_K ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ).

Proof.

Let B,BK(v0)𝐵superscript𝐵𝐾subscript𝑣0B,B^{\prime}\in K(v_{0})italic_B , italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_K ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) and let B=Bksuperscript𝐵superscript𝐵𝑘B^{\prime}=B^{k}italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_B start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT for some kK𝑘𝐾k\in Kitalic_k ∈ italic_K. Let wW𝑤𝑊w\in Witalic_w ∈ italic_W. Then B(w)={B1k:B1B(w)}superscript𝐵𝑤conditional-setsuperscriptsubscript𝐵1𝑘subscript𝐵1𝐵𝑤B^{\prime}(w)=\{B_{1}^{k}:B_{1}\in B(w)\}italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_w ) = { italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT : italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∈ italic_B ( italic_w ) }. Thus K(v)B(w)¯𝐾𝑣¯superscript𝐵𝑤K(v)\cap\overline{B^{\prime}(w)}\neq\emptysetitalic_K ( italic_v ) ∩ ¯ start_ARG italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_w ) end_ARG ≠ ∅ if and only if K(v)B(w)¯𝐾𝑣¯𝐵𝑤K(v)\cap\overline{B(w)}\neq\emptysetitalic_K ( italic_v ) ∩ ¯ start_ARG italic_B ( italic_w ) end_ARG ≠ ∅.

Q.E.D.

We now have the following combinatorial interpretation of Wv0(v)subscript𝑊subscript𝑣0𝑣W_{v_{0}}(v)italic_W start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ).

Lemma 4.2.

For any v0V0subscript𝑣0subscript𝑉0v_{0}\in V_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and vV𝑣𝑉v\in Vitalic_v ∈ italic_V, one has

Wv0(v)={wW:vm(w)v0}.subscript𝑊subscript𝑣0𝑣conditional-set𝑤𝑊𝑣𝑚𝑤subscript𝑣0W_{v_{0}}(v)=\{w\in W:\;v\leq m(w)\!\cdot\!v_{0}\}.italic_W start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) = { italic_w ∈ italic_W : italic_v ≤ italic_m ( italic_w ) ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT } .
Proof.

Let wW𝑤𝑊w\in Witalic_w ∈ italic_W. By Lemma 3.4 and by the definition of the Bruhat order on V𝑉Vitalic_V, vm(w)v0𝑣𝑚𝑤subscript𝑣0v\leq m(w)\!\cdot\!v_{0}italic_v ≤ italic_m ( italic_w ) ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT if and only if K(v)K(m(w)v0)¯=B(w)¯K𝐾𝑣¯𝐾𝑚𝑤subscript𝑣0¯𝐵𝑤𝐾K(v)\subset\overline{K(m(w)\!\cdot\!v_{0})}=\overline{B(w)}\cdot Kitalic_K ( italic_v ) ⊂ ¯ start_ARG italic_K ( italic_m ( italic_w ) ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) end_ARG = ¯ start_ARG italic_B ( italic_w ) end_ARG ⋅ italic_K, which is equivalent to K(v)B(w)¯𝐾𝑣¯𝐵𝑤K(v)\cap\overline{B(w)}\neq\emptysetitalic_K ( italic_v ) ∩ ¯ start_ARG italic_B ( italic_w ) end_ARG ≠ ∅.

Q.E.D.

4.2. Properties of Wv0(v)subscript𝑊subscript𝑣0𝑣W_{v_{0}}(v)italic_W start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v )

Recall from §§\lx@sectionsign§2.2 that for v0V0subscript𝑣0subscript𝑉0v_{0}\in V_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and vV𝑣𝑉v\in Vitalic_v ∈ italic_V, Yv0(v)subscript𝑌subscript𝑣0𝑣Y_{v_{0}}(v)italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) is the set of minimal elements in Wv0(v)subscript𝑊subscript𝑣0𝑣W_{v_{0}}(v)italic_W start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) with respect to the Bruhat order on W𝑊Witalic_W.

Lemma 4.3.

Let v0V0subscript𝑣0subscript𝑉0v_{0}\in V_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, vV𝑣𝑉v\in Vitalic_v ∈ italic_V, and wW𝑤𝑊w\in Witalic_w ∈ italic_W. Then wWv0(v)𝑤subscript𝑊subscript𝑣0𝑣w\in W_{v_{0}}(v)italic_w ∈ italic_W start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) if and only if wy𝑤𝑦w\geq yitalic_w ≥ italic_y for some yYv0(v)𝑦subscript𝑌subscript𝑣0𝑣y\in Y_{v_{0}}(v)italic_y ∈ italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ).

Proof.

It follows from the definition of Yv0(v)subscript𝑌subscript𝑣0𝑣Y_{v_{0}}(v)italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) that wy𝑤𝑦w\geq yitalic_w ≥ italic_y for some yYv0(v)𝑦subscript𝑌subscript𝑣0𝑣y\in Y_{v_{0}}(v)italic_y ∈ italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ). Conversely, assume that wy𝑤𝑦w\geq yitalic_w ≥ italic_y for some yYv0(v)𝑦subscript𝑌subscript𝑣0𝑣y\in Y_{v_{0}}(v)italic_y ∈ italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ). Then K(v)B(y)¯𝐾𝑣¯𝐵𝑦K(v)\cap\overline{B(y)}\neq\emptysetitalic_K ( italic_v ) ∩ ¯ start_ARG italic_B ( italic_y ) end_ARG ≠ ∅ and B(y)¯B(w)¯¯𝐵𝑦¯𝐵𝑤\overline{B(y)}\subset\overline{B(w)}¯ start_ARG italic_B ( italic_y ) end_ARG ⊂ ¯ start_ARG italic_B ( italic_w ) end_ARG. Thus K(v)B(w)¯𝐾𝑣¯𝐵𝑤K(v)\cap\overline{B(w)}\neq\emptysetitalic_K ( italic_v ) ∩ ¯ start_ARG italic_B ( italic_w ) end_ARG ≠ ∅.

Q.E.D.

Lemma 4.4.

For v0V0subscript𝑣0subscript𝑉0v_{0}\in V_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and vV𝑣𝑉v\in Vitalic_v ∈ italic_V, one has Wv0(v)=Wsubscript𝑊subscript𝑣0𝑣𝑊W_{v_{0}}(v)=Witalic_W start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) = italic_W if and only if v=v0𝑣subscript𝑣0v=v_{0}italic_v = italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT.

Proof.

Let 1111 be the identity element of W𝑊Witalic_W. It is clear that 1Wv0(v0)1subscript𝑊subscript𝑣0subscript𝑣01\in W_{v_{0}}(v_{0})1 ∈ italic_W start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ), so Wv0(v0)=Wsubscript𝑊subscript𝑣0subscript𝑣0𝑊W_{v_{0}}(v_{0})=Witalic_W start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) = italic_W by Lemma 4.3. Assume that vV𝑣𝑉v\in Vitalic_v ∈ italic_V is such that Wv0(v)=Wsubscript𝑊subscript𝑣0𝑣𝑊W_{v_{0}}(v)=Witalic_W start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) = italic_W. Then 1Wv0(v)1subscript𝑊subscript𝑣0𝑣1\in W_{v_{0}}(v)1 ∈ italic_W start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ), so vv0𝑣subscript𝑣0v\leq v_{0}italic_v ≤ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT. Since K(v0)𝐾subscript𝑣0K(v_{0})\subset{\mathcal{B}}italic_K ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) ⊂ caligraphic_B is closed, we have v=v0𝑣subscript𝑣0v=v_{0}italic_v = italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT.

Q.E.D.

4.3. The set Wv0(v)superscriptsubscript𝑊subscript𝑣0𝑣W_{v_{0}}^{\prime}(v)italic_W start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_v )

Fix v0V0subscript𝑣0subscript𝑉0v_{0}\in V_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and let BK(v0)𝐵𝐾subscript𝑣0B\in K(v_{0})italic_B ∈ italic_K ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ). For vV𝑣𝑉v\in Vitalic_v ∈ italic_V, let

(4.1) Wv0(v)={wW:K(v)B(w)}.superscriptsubscript𝑊subscript𝑣0𝑣conditional-set𝑤𝑊𝐾𝑣𝐵𝑤W_{v_{0}}^{\prime}(v)=\{w\in W:\;K(v)\cap B(w)\neq\emptyset\}.italic_W start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_v ) = { italic_w ∈ italic_W : italic_K ( italic_v ) ∩ italic_B ( italic_w ) ≠ ∅ } .

By the proof of Lemma 4.1, Wv0(v)superscriptsubscript𝑊subscript𝑣0𝑣W_{v_{0}}^{\prime}(v)italic_W start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_v ) is independent of the choice of B𝐵Bitalic_B in K(v0)𝐾subscript𝑣0K(v_{0})italic_K ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ). One also sees from the definition of B(w)𝐵𝑤B(w)italic_B ( italic_w ) in (3.5) that for any v0V0subscript𝑣0subscript𝑉0v_{0}\in V_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and vV𝑣𝑉v\in Vitalic_v ∈ italic_V,

(4.2) Wv0(v)={wW:w=p(B,B)for someBK(v),BK(v0)}.superscriptsubscript𝑊subscript𝑣0𝑣conditional-set𝑤𝑊formulae-sequence𝑤𝑝superscript𝐵𝐵for somesuperscript𝐵𝐾𝑣𝐵𝐾subscript𝑣0W_{v_{0}}^{\prime}(v)=\{w\in W:\;w=p(B^{\prime},B)\;\mbox{for some}\;B^{\prime% }\in K(v),\,B\in K(v_{0})\}.italic_W start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_v ) = { italic_w ∈ italic_W : italic_w = italic_p ( italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_B ) for some italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_K ( italic_v ) , italic_B ∈ italic_K ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) } .

The following Proposition 4.5 expresses Yv0(v)subscript𝑌subscript𝑣0𝑣Y_{v_{0}}(v)italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) and Zv0(v)subscript𝑍subscript𝑣0𝑣Z_{v_{0}}(v)italic_Z start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) in terms of Wv0(v)superscriptsubscript𝑊subscript𝑣0𝑣W_{v_{0}}^{\prime}(v)italic_W start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_v ).

Proposition 4.5.

For any v0V0subscript𝑣0subscript𝑉0v_{0}\in V_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and vV𝑣𝑉v\in Vitalic_v ∈ italic_V, one has

(4.3) Yv0(v)Wv0(v)Wv0(v).subscript𝑌subscript𝑣0𝑣superscriptsubscript𝑊subscript𝑣0𝑣subscript𝑊subscript𝑣0𝑣Y_{v_{0}}(v)\subset W_{v_{0}}^{\prime}(v)\subset W_{v_{0}}(v).italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) ⊂ italic_W start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_v ) ⊂ italic_W start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) .

Consequently, Yv0(v)=min(Wv0(v))subscript𝑌subscript𝑣0𝑣superscriptsubscript𝑊subscript𝑣0normal-′𝑣Y_{v_{0}}(v)=\min(W_{v_{0}}^{\prime}(v))italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) = roman_min ( italic_W start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_v ) ) and Zv0(v)=minl(Wv0(v))subscript𝑍subscript𝑣0𝑣subscript𝑙superscriptsubscript𝑊subscript𝑣0normal-′𝑣Z_{v_{0}}(v)=\min_{l}(W_{v_{0}}^{\prime}(v))italic_Z start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) = roman_min start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ( italic_W start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_v ) ).

Proof.

Clearly Wv0(v)Wv0(v)superscriptsubscript𝑊subscript𝑣0𝑣subscript𝑊subscript𝑣0𝑣W_{v_{0}}^{\prime}(v)\subset W_{v_{0}}(v)italic_W start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_v ) ⊂ italic_W start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ). Let wYv0(v)𝑤subscript𝑌subscript𝑣0𝑣w\in Y_{v_{0}}(v)italic_w ∈ italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ). Then K(v)B(w)¯𝐾𝑣¯𝐵𝑤K(v)\cap\overline{B(w)}\neq\emptysetitalic_K ( italic_v ) ∩ ¯ start_ARG italic_B ( italic_w ) end_ARG ≠ ∅ and K(v)B(w)=𝐾𝑣𝐵superscript𝑤K(v)\cap B(w^{\prime})=\emptysetitalic_K ( italic_v ) ∩ italic_B ( italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) = ∅ for any wWsuperscript𝑤𝑊w^{\prime}\in Witalic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_W such that w<wsuperscript𝑤𝑤w^{\prime}<witalic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT < italic_w. Thus K(v)B(w)𝐾𝑣𝐵𝑤K(v)\cap B(w)\neq\emptysetitalic_K ( italic_v ) ∩ italic_B ( italic_w ) ≠ ∅ and wWv0(v)𝑤superscriptsubscript𝑊subscript𝑣0𝑣w\in W_{v_{0}}^{\prime}(v)italic_w ∈ italic_W start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_v ). By Lemma 3.1, Yv0(v)=min(Wv0(v))subscript𝑌subscript𝑣0𝑣superscriptsubscript𝑊subscript𝑣0𝑣Y_{v_{0}}(v)=\min(W_{v_{0}}^{\prime}(v))italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) = roman_min ( italic_W start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_v ) ), and Zv0(v)=minl(Wv0(v))subscript𝑍subscript𝑣0𝑣subscript𝑙superscriptsubscript𝑊subscript𝑣0𝑣Z_{v_{0}}(v)=\min_{l}(W_{v_{0}}^{\prime}(v))italic_Z start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) = roman_min start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ( italic_W start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_v ) ).

Q.E.D.

Example 4.6.

Let G~=G×G~𝐺𝐺𝐺\tilde{G}=G\times G~ start_ARG italic_G end_ARG = italic_G × italic_G and let

θ~:G~G~:θ~(g1,g2)=(g2,g1),(g1,g2)G~.:~𝜃~𝐺~𝐺:formulae-sequence~𝜃subscript𝑔1subscript𝑔2subscript𝑔2subscript𝑔1subscript𝑔1subscript𝑔2~𝐺\tilde{\theta}:\;\;\tilde{G}\longrightarrow\tilde{G}:\;\;\;\tilde{\theta}(g_{1% },g_{2})=(g_{2},g_{1}),\hskip 14.454pt(g_{1},g_{2})\in\tilde{G}.~ start_ARG italic_θ end_ARG : ~ start_ARG italic_G end_ARG ⟶ ~ start_ARG italic_G end_ARG : ~ start_ARG italic_θ end_ARG ( italic_g start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_g start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) = ( italic_g start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_g start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) , ( italic_g start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_g start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ∈ ~ start_ARG italic_G end_ARG .

Then the fixed point subgroup K~~𝐾\tilde{K}~ start_ARG italic_K end_ARG of θ~~𝜃\tilde{\theta}~ start_ARG italic_θ end_ARG in G~~𝐺\tilde{G}~ start_ARG italic_G end_ARG is K~={(g,g):gG}~𝐾conditional-set𝑔𝑔𝑔𝐺\tilde{K}=\{(g,g):g\in G\}~ start_ARG italic_K end_ARG = { ( italic_g , italic_g ) : italic_g ∈ italic_G }, so the set V~~𝑉\tilde{V}~ start_ARG italic_V end_ARG of K~~𝐾\tilde{K}~ start_ARG italic_K end_ARG-orbits in ~=×~\tilde{{\mathcal{B}}}={\mathcal{B}}\times{\mathcal{B}}~ start_ARG caligraphic_B end_ARG = caligraphic_B × caligraphic_B is W𝑊Witalic_W. For wW𝑤𝑊w\in Witalic_w ∈ italic_W, let K~(w)=𝒪(w)~𝐾𝑤𝒪𝑤\tilde{K}(w)={\mathcal{O}}(w)~ start_ARG italic_K end_ARG ( italic_w ) = caligraphic_O ( italic_w ), where we recall that 𝒪(w)𝒪𝑤{\mathcal{O}}(w)caligraphic_O ( italic_w ) is the G𝐺Gitalic_G-orbit in ~~\tilde{{\mathcal{B}}}~ start_ARG caligraphic_B end_ARG for the diagonal action. Then the only v~0V~=Wsubscript~𝑣0~𝑉𝑊\tilde{v}_{0}\in\tilde{V}=W~ start_ARG italic_v end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ ~ start_ARG italic_V end_ARG = italic_W such that K~(v~0)~𝐾subscript~𝑣0\tilde{K}(\tilde{v}_{0})~ start_ARG italic_K end_ARG ( ~ start_ARG italic_v end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) is closed in ~~\tilde{{\mathcal{B}}}~ start_ARG caligraphic_B end_ARG is v~0=1subscript~𝑣01\tilde{v}_{0}=1~ start_ARG italic_v end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = 1, the identity element of W𝑊Witalic_W. Let W~=W×W~𝑊𝑊𝑊\tilde{W}=W\times W~ start_ARG italic_W end_ARG = italic_W × italic_W, and for wW=V~𝑤𝑊~𝑉w\in W=\tilde{V}italic_w ∈ italic_W = ~ start_ARG italic_V end_ARG, let W~v~0(w)W~v~0(w)W~superscriptsubscript~𝑊subscript~𝑣0𝑤subscript~𝑊subscript~𝑣0𝑤~𝑊\tilde{W}_{\tilde{v}_{0}}^{\prime}(w)\subset\tilde{W}_{\tilde{v}_{0}}(w)% \subset\tilde{W}~ start_ARG italic_W end_ARG start_POSTSUBSCRIPT ~ start_ARG italic_v end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_w ) ⊂ ~ start_ARG italic_W end_ARG start_POSTSUBSCRIPT ~ start_ARG italic_v end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_w ) ⊂ ~ start_ARG italic_W end_ARG be defined as in (2.2) and (4.1) but for the pair (G~,θ~)~𝐺~𝜃(\tilde{G},\tilde{\theta})( ~ start_ARG italic_G end_ARG , ~ start_ARG italic_θ end_ARG ). Then it is easy to see that, for any wW=V~𝑤𝑊~𝑉w\in W=\tilde{V}italic_w ∈ italic_W = ~ start_ARG italic_V end_ARG,

W~v~0(w)={(w1,w2)W×W:ww1w2-1},subscript~𝑊subscript~𝑣0𝑤conditional-setsubscript𝑤1subscript𝑤2𝑊𝑊𝑤subscript𝑤1superscriptsubscript𝑤21\tilde{W}_{\tilde{v}_{0}}(w)=\{(w_{1},w_{2})\in W\times W:\;w\leq w_{1}\ast w_% {2}^{-1}\},~ start_ARG italic_W end_ARG start_POSTSUBSCRIPT ~ start_ARG italic_v end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_w ) = { ( italic_w start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ∈ italic_W × italic_W : italic_w ≤ italic_w start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∗ italic_w start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT } ,

where recall from §§\lx@sectionsign§3.6 that \ast is the monoidal product on W𝑊Witalic_W, while

W~v~0(w)={(w1,w2)W×W:BwBBw1Bw2-1B},superscriptsubscript~𝑊subscript~𝑣0𝑤conditional-setsubscript𝑤1subscript𝑤2𝑊𝑊𝐵𝑤𝐵𝐵subscript𝑤1𝐵superscriptsubscript𝑤21𝐵\tilde{W}_{\tilde{v}_{0}}^{\prime}(w)=\{(w_{1},w_{2})\in W\times W:\;BwB% \subset Bw_{1}Bw_{2}^{-1}B\},~ start_ARG italic_W end_ARG start_POSTSUBSCRIPT ~ start_ARG italic_v end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_w ) = { ( italic_w start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ∈ italic_W × italic_W : italic_B italic_w italic_B ⊂ italic_B italic_w start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_B italic_w start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_B } ,

where B𝐵Bitalic_B is any Borel subgroup of G𝐺Gitalic_G. The set W~v~0(w)superscriptsubscript~𝑊subscript~𝑣0𝑤\tilde{W}_{\tilde{v}_{0}}^{\prime}(w)~ start_ARG italic_W end_ARG start_POSTSUBSCRIPT ~ start_ARG italic_v end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_w ) can be computed by choosing any reduced word for w1subscript𝑤1w_{1}italic_w start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and using inductively the fact that, for sS𝑠𝑆s\in Sitalic_s ∈ italic_S and uW𝑢𝑊u\in Witalic_u ∈ italic_W, BsBuB=BsuB𝐵𝑠𝐵𝑢𝐵𝐵𝑠𝑢𝐵BsBuB=BsuBitalic_B italic_s italic_B italic_u italic_B = italic_B italic_s italic_u italic_B if su>u𝑠𝑢𝑢su>uitalic_s italic_u > italic_u and BsBuB=BsuBBuB𝐵𝑠𝐵𝑢𝐵𝐵𝑠𝑢𝐵𝐵𝑢𝐵BsBuB=BsuB\cup BuBitalic_B italic_s italic_B italic_u italic_B = italic_B italic_s italic_u italic_B ∪ italic_B italic_u italic_B if su<u𝑠𝑢𝑢su<uitalic_s italic_u < italic_u. See [6, Remarques 3.19]. Moreover, for any wW𝑤𝑊w\in Witalic_w ∈ italic_W,

min(W~v~0(w))minsubscript~𝑊subscript~𝑣0𝑤\displaystyle{\rm min}(\tilde{W}_{\tilde{v}_{0}}(w))roman_min ( ~ start_ARG italic_W end_ARG start_POSTSUBSCRIPT ~ start_ARG italic_v end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_w ) ) =min(W~v~0(w))absentminsuperscriptsubscript~𝑊subscript~𝑣0𝑤\displaystyle={\rm min}(\tilde{W}_{\tilde{v}_{0}}^{\prime}(w))= roman_min ( ~ start_ARG italic_W end_ARG start_POSTSUBSCRIPT ~ start_ARG italic_v end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_w ) )
={(w1,w2)W×W:w=w1w2-1,l(w)=l(w1)+l(w2)},absentconditional-setsubscript𝑤1subscript𝑤2𝑊𝑊formulae-sequence𝑤subscript𝑤1superscriptsubscript𝑤21𝑙𝑤𝑙subscript𝑤1𝑙subscript𝑤2\displaystyle=\{(w_{1},w_{2})\in W\times W:\;w=w_{1}w_{2}^{-1},\;l(w)=l(w_{1})% +l(w_{2})\},= { ( italic_w start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ∈ italic_W × italic_W : italic_w = italic_w start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_w start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT , italic_l ( italic_w ) = italic_l ( italic_w start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) + italic_l ( italic_w start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) } ,

and all elements in min(W~v~0(w))minsubscript~𝑊subscript~𝑣0𝑤{\rm min}(\tilde{W}_{\tilde{v}_{0}}(w))roman_min ( ~ start_ARG italic_W end_ARG start_POSTSUBSCRIPT ~ start_ARG italic_v end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_w ) ) have the same length, namely l(w)𝑙𝑤l(w)italic_l ( italic_w ).

4.4. The element m(w)v0𝑚𝑤subscript𝑣0m(w)\!\cdot\!v_{0}italic_m ( italic_w ) ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT

Fix v0V0subscript𝑣0subscript𝑉0v_{0}\in V_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and let BK(v0)𝐵𝐾subscript𝑣0B\in K(v_{0})italic_B ∈ italic_K ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ). For wW𝑤𝑊w\in Witalic_w ∈ italic_W, let

(4.4) Vw={vV:K(v)B(w)}.subscript𝑉𝑤conditional-set𝑣𝑉𝐾𝑣𝐵𝑤V_{w}=\{v\in V:\;K(v)\cap B(w)\neq\emptyset\}.italic_V start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT = { italic_v ∈ italic_V : italic_K ( italic_v ) ∩ italic_B ( italic_w ) ≠ ∅ } .

Then Vwsubscript𝑉𝑤V_{w}italic_V start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT depends only on w𝑤witalic_w and not on the choice of BK(v0)𝐵𝐾subscript𝑣0B\in K(v_{0})italic_B ∈ italic_K ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ).

Proposition 4.7.

Let v0V0subscript𝑣0subscript𝑉0v_{0}\in V_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and let BK(v0)𝐵𝐾subscript𝑣0B\in K(v_{0})italic_B ∈ italic_K ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ). Then for every wW𝑤𝑊w\in Witalic_w ∈ italic_W,

1) B(w)K(m(w)v0)𝐵𝑤𝐾normal-⋅𝑚𝑤subscript𝑣0B(w)\cap K(m(w)\!\cdot\!v_{0})italic_B ( italic_w ) ∩ italic_K ( italic_m ( italic_w ) ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) is dense in B(w)𝐵𝑤B(w)italic_B ( italic_w ),

2) m(w)v0normal-⋅𝑚𝑤subscript𝑣0m(w)\!\cdot\!v_{0}italic_m ( italic_w ) ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT is the unique maximal element in Vwsubscript𝑉𝑤V_{w}italic_V start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT with respect to the Bruhat order on V𝑉Vitalic_V.

Proof.

Since B(w)𝐵𝑤B(w)italic_B ( italic_w ) is irreducible and since =vVK(v)subscriptsquare-union𝑣𝑉𝐾𝑣{\mathcal{B}}=\bigsqcup_{v\in V}K(v)caligraphic_B = ⊔ start_POSTSUBSCRIPT italic_v ∈ italic_V end_POSTSUBSCRIPT italic_K ( italic_v ), there exists a unique vwVsubscript𝑣𝑤𝑉v_{w}\in Vitalic_v start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT ∈ italic_V such that B(w)K(vw)𝐵𝑤𝐾subscript𝑣𝑤B(w)\cap K(v_{w})italic_B ( italic_w ) ∩ italic_K ( italic_v start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT ) is dense in B(w)𝐵𝑤B(w)italic_B ( italic_w ). Moreover, if vVw𝑣subscript𝑉𝑤v\in V_{w}italic_v ∈ italic_V start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT, then

B(w)K(v)B(w)B(w)¯=B(w)K(vw)¯K(vw)¯,𝐵𝑤𝐾𝑣𝐵𝑤¯𝐵𝑤¯𝐵𝑤𝐾subscript𝑣𝑤¯𝐾subscript𝑣𝑤\emptyset\neq B(w)\cap K(v)\subset B(w)\subset\overline{B(w)}=\overline{B(w)% \cap K(v_{w})}\subset\overline{K(v_{w})},∅ ≠ italic_B ( italic_w ) ∩ italic_K ( italic_v ) ⊂ italic_B ( italic_w ) ⊂ ¯ start_ARG italic_B ( italic_w ) end_ARG = ¯ start_ARG italic_B ( italic_w ) ∩ italic_K ( italic_v start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT ) end_ARG ⊂ ¯ start_ARG italic_K ( italic_v start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT ) end_ARG ,

and so vvw𝑣subscript𝑣𝑤v\leq v_{w}italic_v ≤ italic_v start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT. This shows that vwsubscript𝑣𝑤v_{w}italic_v start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT is the unique maximal element in Vwsubscript𝑉𝑤V_{w}italic_V start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT with respect to the Bruhat order on V𝑉Vitalic_V. It remains to show that vw=m(w)v0subscript𝑣𝑤𝑚𝑤subscript𝑣0v_{w}=m(w)\!\cdot\!v_{0}italic_v start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT = italic_m ( italic_w ) ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT.

Since wWv0(vw)𝑤subscript𝑊subscript𝑣0subscript𝑣𝑤w\in W_{v_{0}}(v_{w})italic_w ∈ italic_W start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT ), vwm(w)v0subscript𝑣𝑤𝑚𝑤subscript𝑣0v_{w}\leq m(w)\!\cdot\!v_{0}italic_v start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT ≤ italic_m ( italic_w ) ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT. By Lemma 3.4, K(m(w)v0)¯=B(w)K¯¯𝐾𝑚𝑤subscript𝑣0¯𝐵𝑤𝐾\overline{K(m(w)\!\cdot\!v_{0})}=\overline{B(w)\cdot K}¯ start_ARG italic_K ( italic_m ( italic_w ) ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) end_ARG = ¯ start_ARG italic_B ( italic_w ) ⋅ italic_K end_ARG, so K(m(w)v0)(B(w)K)𝐾𝑚𝑤subscript𝑣0𝐵𝑤𝐾K(m(w)\!\cdot\!v_{0})\cap(B(w)\cdot K)\neq\emptysetitalic_K ( italic_m ( italic_w ) ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) ∩ ( italic_B ( italic_w ) ⋅ italic_K ) ≠ ∅, and hence K(m(w)v0)B(w)𝐾𝑚𝑤subscript𝑣0𝐵𝑤K(m(w)\!\cdot\!v_{0})\cap B(w)\neq\emptysetitalic_K ( italic_m ( italic_w ) ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) ∩ italic_B ( italic_w ) ≠ ∅. Thus m(w)v0Vw𝑚𝑤subscript𝑣0subscript𝑉𝑤m(w)\!\cdot\!v_{0}\in V_{w}italic_m ( italic_w ) ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ italic_V start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT and m(w)v0vw𝑚𝑤subscript𝑣0subscript𝑣𝑤m(w)\!\cdot\!v_{0}\leq v_{w}italic_m ( italic_w ) ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ≤ italic_v start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT. Hence vw=m(w)v0subscript𝑣𝑤𝑚𝑤subscript𝑣0v_{w}=m(w)\!\cdot\!v_{0}italic_v start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT = italic_m ( italic_w ) ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT.

Q.E.D.

5. Intersections of G0subscript𝐺0G_{0}italic_G start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT-orbits and B𝐵Bitalic_B-orbits

In this section, we assume that 𝐤=𝐤{\bf k}={\mathbb{C}}bold_k = blackboard_C and that G0={gG:τ(g)=g}subscript𝐺0conditional-set𝑔𝐺𝜏𝑔𝑔G_{0}=\{g\in G:\tau(g)=g\}italic_G start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = { italic_g ∈ italic_G : italic_τ ( italic_g ) = italic_g } is a real form of G𝐺Gitalic_G, where τ𝜏\tauitalic_τ is an anti-holomorphic involution on G𝐺Gitalic_G. Let σ𝜎\sigmaitalic_σ be a Cartan involution of G𝐺Gitalic_G commuting with τ𝜏\tauitalic_τ, and let θ=στ𝜃𝜎𝜏\theta=\sigma\tauitalic_θ = italic_σ italic_τ. Let K=Gθ𝐾superscript𝐺𝜃K=G^{\theta}italic_K = italic_G start_POSTSUPERSCRIPT italic_θ end_POSTSUPERSCRIPT and K0=G0Ksubscript𝐾0subscript𝐺0𝐾K_{0}=G_{0}\cap Kitalic_K start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_G start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∩ italic_K. Then K0subscript𝐾0K_{0}italic_K start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT is a maximal compact subgroup of G0subscript𝐺0G_{0}italic_G start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and K𝐾Kitalic_K is a complexification of K0subscript𝐾0K_{0}italic_K start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT.

If X𝑋Xitalic_X is a subset of {\mathcal{B}}caligraphic_B, X¯¯𝑋\overline{X}¯ start_ARG italic_X end_ARG in this section will denote the closure of X𝑋Xitalic_X in {\mathcal{B}}caligraphic_B in the classical topology. If X=K(v)𝑋𝐾𝑣X=K(v)italic_X = italic_K ( italic_v ) or X=B(w)𝑋𝐵𝑤X=B(w)italic_X = italic_B ( italic_w ), where vV𝑣𝑉v\in Vitalic_v ∈ italic_V, B𝐵Bitalic_B is any Borel subgroup of G𝐺Gitalic_G, and wW𝑤𝑊w\in Witalic_w ∈ italic_W, the closure of X𝑋Xitalic_X in the classical topology coincides with that in the Zariski topology.

5.1. The Matsuki duality

By Matsuki duality [8, 10] [13, §§\lx@sectionsign§6], for each vV𝑣𝑉v\in Vitalic_v ∈ italic_V, there exists a unique G0subscript𝐺0G_{0}italic_G start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT-orbit G0(v)subscript𝐺0𝑣G_{0}(v)italic_G start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_v ) in {\mathcal{B}}caligraphic_B such that G0(v)K(v)subscript𝐺0𝑣𝐾𝑣G_{0}(v)\cap K(v)italic_G start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_v ) ∩ italic_K ( italic_v ) is a single K0subscript𝐾0K_{0}italic_K start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT-orbit. The map vG0(v)maps-to𝑣subscript𝐺0𝑣v\mapsto G_{0}(v)italic_v ↦ italic_G start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_v ) gives an identification of the set V=/K𝑉𝐾V={\mathcal{B}}/Kitalic_V = caligraphic_B / italic_K of K𝐾Kitalic_K-orbits in {\mathcal{B}}caligraphic_B with the set /G0subscript𝐺0{\mathcal{B}}/G_{0}caligraphic_B / italic_G start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT of G0subscript𝐺0G_{0}italic_G start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT-orbits in {\mathcal{B}}caligraphic_B. For vV𝑣𝑉v\in Vitalic_v ∈ italic_V, let K0(v)=G0(v)K(v).subscript𝐾0𝑣subscript𝐺0𝑣𝐾𝑣K_{0}(v)=G_{0}(v)\cap K(v).italic_K start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_v ) = italic_G start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_v ) ∩ italic_K ( italic_v ) .

The following Proposition 5.1 is proved in [10, Corollary 1.4] [13, Theorem 6.4.5].

Proposition 5.1.

For v,vV𝑣superscript𝑣normal-′𝑉v,v^{\prime}\in Vitalic_v , italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_V, one has

vv  𝑖𝑓𝑓  G0(v)G0(v)¯  𝑖𝑓𝑓  G0(v)K(v).formulae-sequence𝑣superscript𝑣𝑖𝑓𝑓formulae-sequencesubscript𝐺0superscript𝑣¯subscript𝐺0𝑣𝑖𝑓𝑓subscript𝐺0𝑣𝐾superscript𝑣v\leq v^{\prime}\hskip 14.454pt\mbox{iff}\hskip 14.454ptG_{0}(v^{\prime})% \subset\overline{G_{0}(v)}\hskip 14.454pt\mbox{iff}\hskip 14.454ptG_{0}(v)\cap K% (v^{\prime})\neq\emptyset.italic_v ≤ italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT iff italic_G start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ⊂ ¯ start_ARG italic_G start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_v ) end_ARG iff italic_G start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_v ) ∩ italic_K ( italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ≠ ∅ .

Let v0V0subscript𝑣0subscript𝑉0v_{0}\in V_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and fix BK(v0)𝐵𝐾subscript𝑣0B\in K(v_{0})italic_B ∈ italic_K ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ). Recall the map qB:G:gBg:subscript𝑞𝐵𝐺:maps-to𝑔superscript𝐵𝑔q_{B}:G\rightarrow{\mathcal{B}}:g\mapsto B^{g}italic_q start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT : italic_G → caligraphic_B : italic_g ↦ italic_B start_POSTSUPERSCRIPT italic_g end_POSTSUPERSCRIPT for gG𝑔𝐺g\in Gitalic_g ∈ italic_G. For vV𝑣𝑉v\in Vitalic_v ∈ italic_V, let

BvG0=qB-1(G0(v)),BvK0=qB-1(K0(v)),formulae-sequence𝐵𝑣subscript𝐺0superscriptsubscript𝑞𝐵1subscript𝐺0𝑣𝐵𝑣subscript𝐾0superscriptsubscript𝑞𝐵1subscript𝐾0𝑣BvG_{0}=q_{B}^{-1}(G_{0}(v)),\hskip 14.454ptBvK_{0}=q_{B}^{-1}(K_{0}(v)),italic_B italic_v italic_G start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_q start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_G start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_v ) ) , italic_B italic_v italic_K start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_q start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_K start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_v ) ) ,

and recall that BvK=qB-1(K(v))𝐵𝑣𝐾superscriptsubscript𝑞𝐵1𝐾𝑣BvK=q_{B}^{-1}(K(v))italic_B italic_v italic_K = italic_q start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_K ( italic_v ) ). Let BK=KBsubscript𝐵𝐾𝐾𝐵B_{K}=K\cap Bitalic_B start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT = italic_K ∩ italic_B.

Lemma 5.2.

1) K=K0BK𝐾subscript𝐾0subscript𝐵𝐾K=K_{0}B_{K}italic_K = italic_K start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT and G0K=G0BKsubscript𝐺0𝐾subscript𝐺0subscript𝐵𝐾G_{0}K=G_{0}B_{K}italic_G start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_K = italic_G start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT.

2) For any vV𝑣𝑉v\in Vitalic_v ∈ italic_V, one has BvG0K=vV,vvBvK𝐵𝑣subscript𝐺0𝐾subscriptsquare-unionformulae-sequencesuperscript𝑣normal-′𝑉superscript𝑣normal-′𝑣𝐵superscript𝑣normal-′𝐾BvG_{0}K=\bigsqcup_{v^{\prime}\in V,v^{\prime}\geq v}Bv^{\prime}Kitalic_B italic_v italic_G start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_K = ⊔ start_POSTSUBSCRIPT italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_V , italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≥ italic_v end_POSTSUBSCRIPT italic_B italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_K.

Proof.

1) Let U={gG:σ(g)=g}𝑈conditional-set𝑔𝐺𝜎𝑔𝑔U=\{g\in G:\sigma(g)=g\}italic_U = { italic_g ∈ italic_G : italic_σ ( italic_g ) = italic_g } and H=Bσ(B)𝐻𝐵𝜎𝐵H=B\cap\sigma(B)italic_H = italic_B ∩ italic_σ ( italic_B ). Then U𝑈Uitalic_U is a compact real form of G𝐺Gitalic_G and H𝐻Hitalic_H a maximal torus of G𝐺Gitalic_G. Let A={hH:σ(h)=h-1}𝐴conditional-set𝐻𝜎superscript1A=\{h\in H:\sigma(h)=h^{-1}\}italic_A = { italic_h ∈ italic_H : italic_σ ( italic_h ) = italic_h start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT } and let N𝑁Nitalic_N be the uniradical of B𝐵Bitalic_B. Then one has the Iwasawa decomposition G=UAN𝐺𝑈𝐴𝑁G=UANitalic_G = italic_U italic_A italic_N of G𝐺Gitalic_G, and U𝑈Uitalic_U, A𝐴Aitalic_A, and N𝑁Nitalic_N are all θ𝜃\thetaitalic_θ-invariant. Let kK𝑘𝐾k\in Kitalic_k ∈ italic_K and write k=uan𝑘𝑢𝑎𝑛k=uanitalic_k = italic_u italic_a italic_n with uU,aAformulae-sequence𝑢𝑈𝑎𝐴u\in U,a\in Aitalic_u ∈ italic_U , italic_a ∈ italic_A, and nN𝑛𝑁n\in Nitalic_n ∈ italic_N. It follows from θ(k)=k𝜃𝑘𝑘\theta(k)=kitalic_θ ( italic_k ) = italic_k that uK0𝑢subscript𝐾0u\in K_{0}italic_u ∈ italic_K start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and anBK𝑎𝑛subscript𝐵𝐾an\in B_{K}italic_a italic_n ∈ italic_B start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT. Thus K=K0BK𝐾subscript𝐾0subscript𝐵𝐾K=K_{0}B_{K}italic_K = italic_K start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT. It follows from K0G0subscript𝐾0subscript𝐺0K_{0}\subset G_{0}italic_K start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ⊂ italic_G start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT that G0K=G0BKsubscript𝐺0𝐾subscript𝐺0subscript𝐵𝐾G_{0}K=G_{0}B_{K}italic_G start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_K = italic_G start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT.

2) Let vV𝑣𝑉v\in Vitalic_v ∈ italic_V. Then BvG0K𝐵𝑣subscript𝐺0𝐾BvG_{0}Kitalic_B italic_v italic_G start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_K is a union of (B,K)𝐵𝐾(B,K)( italic_B , italic_K )-double cosets, and for vVsuperscript𝑣𝑉v^{\prime}\in Vitalic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_V,

BvKBvG0K  iff  BvKBvG0,formulae-sequence𝐵superscript𝑣𝐾𝐵𝑣subscript𝐺0𝐾iff𝐵superscript𝑣𝐾𝐵𝑣subscript𝐺0Bv^{\prime}K\subset BvG_{0}K\hskip 14.454pt\mbox{iff}\hskip 14.454ptBv^{\prime% }K\cap BvG_{0}\neq\emptyset,italic_B italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_K ⊂ italic_B italic_v italic_G start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_K iff italic_B italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_K ∩ italic_B italic_v italic_G start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ≠ ∅ ,

which, by Proposition 5.1, is equivalent to vv𝑣superscript𝑣v\leq v^{\prime}italic_v ≤ italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT.

Q.E.D.

5.2. Proof of Theorem 2.1

Let v0V0subscript𝑣0subscript𝑉0v_{0}\in V_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and let BK(v0)𝐵𝐾subscript𝑣0B\in K(v_{0})italic_B ∈ italic_K ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ). Let vV𝑣𝑉v\in Vitalic_v ∈ italic_V and wW𝑤𝑊w\in Witalic_w ∈ italic_W. We must prove that

G0(v)B(w)  iff  vm(w)v0.formulae-sequencesubscript𝐺0𝑣𝐵𝑤iff𝑣𝑚𝑤subscript𝑣0G_{0}(v)\cap B(w)\neq\emptyset\hskip 14.454pt\mbox{iff}\hskip 14.454ptv\leq m(% w)\!\cdot\!v_{0}.italic_G start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_v ) ∩ italic_B ( italic_w ) ≠ ∅ iff italic_v ≤ italic_m ( italic_w ) ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT .

Let BwB=qB-1(B(w))G𝐵𝑤𝐵superscriptsubscript𝑞𝐵1𝐵𝑤𝐺BwB=q_{B}^{-1}(B(w))\subset Gitalic_B italic_w italic_B = italic_q start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_B ( italic_w ) ) ⊂ italic_G. Then, by definition, G0(v)B(w)subscript𝐺0𝑣𝐵𝑤G_{0}(v)\cap B(w)\neq\emptysetitalic_G start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_v ) ∩ italic_B ( italic_w ) ≠ ∅ if and only if (BvG0)(BwB)𝐵𝑣subscript𝐺0𝐵𝑤𝐵(BvG_{0})\cap(BwB)\neq\emptyset( italic_B italic_v italic_G start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) ∩ ( italic_B italic_w italic_B ) ≠ ∅. Since BKBsubscript𝐵𝐾𝐵B_{K}\subset Bitalic_B start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT ⊂ italic_B, this last statement is equivalent to (BvG0BK)(BwB)𝐵𝑣subscript𝐺0subscript𝐵𝐾𝐵𝑤𝐵(BvG_{0}B_{K})\cap(BwB)\neq\emptyset( italic_B italic_v italic_G start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT ) ∩ ( italic_B italic_w italic_B ) ≠ ∅, which, by 1) of Lemma 5.2, is in turn equivalent to (BvG0K)(BwB)𝐵𝑣subscript𝐺0𝐾𝐵𝑤𝐵(BvG_{0}K)\cap(BwB)\neq\emptyset( italic_B italic_v italic_G start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_K ) ∩ ( italic_B italic_w italic_B ) ≠ ∅. By 2) of Lemma 5.2, G0(v)B(w)subscript𝐺0𝑣𝐵𝑤G_{0}(v)\cap B(w)\neq\emptysetitalic_G start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_v ) ∩ italic_B ( italic_w ) ≠ ∅ if and only if vv𝑣superscript𝑣v\leq v^{\prime}italic_v ≤ italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT for some vVwsuperscript𝑣subscript𝑉𝑤v^{\prime}\in V_{w}italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_V start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT, where Vw={vV:K(v)B(w)}subscript𝑉𝑤conditional-setsuperscript𝑣𝑉𝐾superscript𝑣𝐵𝑤V_{w}=\{v^{\prime}\in V:\;K(v^{\prime})\cap B(w)\neq\emptyset\}italic_V start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT = { italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_V : italic_K ( italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ∩ italic_B ( italic_w ) ≠ ∅ }.

By Proposition 4.7, m(w)v0𝑚𝑤subscript𝑣0m(w)\!\cdot\!v_{0}italic_m ( italic_w ) ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT is the unique maximal element in Vwsubscript𝑉𝑤V_{w}italic_V start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT with respect to the Bruhat order on V𝑉Vitalic_V. Thus G0(v)B(w)subscript𝐺0𝑣𝐵𝑤G_{0}(v)\cap B(w)\neq\emptysetitalic_G start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_v ) ∩ italic_B ( italic_w ) ≠ ∅ if and only if vm(w)v0𝑣𝑚𝑤subscript𝑣0v\leq m(w)\!\cdot\!v_{0}italic_v ≤ italic_m ( italic_w ) ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT.

This finishes the proof of Theorem 2.1.

Corollary 5.3.

Let v0V0subscript𝑣0subscript𝑉0v_{0}\in V_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and let BK(v0)𝐵𝐾subscript𝑣0B\in K(v_{0})italic_B ∈ italic_K ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ). Then for any vV𝑣𝑉v\in Vitalic_v ∈ italic_V and wW𝑤𝑊w\in Witalic_w ∈ italic_W, the following are equivalent:

1) G0(v)B(w)subscript𝐺0𝑣𝐵𝑤G_{0}(v)\cap B(w)\neq\emptysetitalic_G start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_v ) ∩ italic_B ( italic_w ) ≠ ∅;

2) G0(v)¯B(w)normal-¯subscript𝐺0𝑣𝐵𝑤\overline{G_{0}(v)}\cap B(w)\neq\emptyset¯ start_ARG italic_G start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_v ) end_ARG ∩ italic_B ( italic_w ) ≠ ∅;

3) G0(v)B(w)¯subscript𝐺0𝑣normal-¯𝐵𝑤G_{0}(v)\cap\overline{B(w)}\neq\emptysetitalic_G start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_v ) ∩ ¯ start_ARG italic_B ( italic_w ) end_ARG ≠ ∅;

4) G0(v)¯B(w)¯normal-¯subscript𝐺0𝑣normal-¯𝐵𝑤\overline{G_{0}(v)}\cap\overline{B(w)}\neq\emptyset¯ start_ARG italic_G start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_v ) end_ARG ∩ ¯ start_ARG italic_B ( italic_w ) end_ARG ≠ ∅;

5) K(v)B(w)¯𝐾𝑣normal-¯𝐵𝑤K(v)\cap\overline{B(w)}\neq\emptysetitalic_K ( italic_v ) ∩ ¯ start_ARG italic_B ( italic_w ) end_ARG ≠ ∅;

6) K0(v)B(w)¯subscript𝐾0𝑣normal-¯𝐵𝑤K_{0}(v)\cap\overline{B(w)}\neq\emptysetitalic_K start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_v ) ∩ ¯ start_ARG italic_B ( italic_w ) end_ARG ≠ ∅.

Proof.

Clearly 1) implies 4). Assume 4). Then there exist v1Vsubscript𝑣1𝑉v_{1}\in Vitalic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∈ italic_V and w1Wsubscript𝑤1𝑊w_{1}\in Witalic_w start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∈ italic_W with v1vsubscript𝑣1𝑣v_{1}\geq vitalic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≥ italic_v and w1wsubscript𝑤1𝑤w_{1}\leq witalic_w start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≤ italic_w, such that G0(v1)B(w1)subscript𝐺0subscript𝑣1𝐵subscript𝑤1G_{0}(v_{1})\cap B(w_{1})\neq\emptysetitalic_G start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ∩ italic_B ( italic_w start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ≠ ∅. By Theorem 2.1, v1m(w1)v0subscript𝑣1𝑚subscript𝑤1subscript𝑣0v_{1}\leq m(w_{1})\!\cdot\!v_{0}italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≤ italic_m ( italic_w start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and by Lemma 3.5, vv1m(w1)v0m(w)v0.𝑣subscript𝑣1𝑚subscript𝑤1subscript𝑣0𝑚𝑤subscript𝑣0v\leq v_{1}\leq m(w_{1})\!\cdot\!v_{0}\leq m(w)\!\cdot\!v_{0}.italic_v ≤ italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≤ italic_m ( italic_w start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ≤ italic_m ( italic_w ) ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT . Thus 1) holds by Theorem 2.1. Hence 1) is equivalent to 4) and consequently also equivalent to 2) and 3).

By Theorem 2.1 and Lemma 4.2, both 1) and 5) are equivalent to vm(w)v0𝑣𝑚𝑤subscript𝑣0v\leq m(w)\!\cdot\!v_{0}italic_v ≤ italic_m ( italic_w ) ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT. Since K=K0BK𝐾subscript𝐾0subscript𝐵𝐾K=K_{0}B_{K}italic_K = italic_K start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT by 1) of Lemma 5.2, 5) and 6) are equivalent.

Q.E.D.

Lemma 5.4.

Let v0V0subscript𝑣0subscript𝑉0v_{0}\in V_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and let BK(v0)𝐵𝐾subscript𝑣0B\in K(v_{0})italic_B ∈ italic_K ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ). Then for any vV𝑣𝑉v\in Vitalic_v ∈ italic_V and wW𝑤𝑊w\in Witalic_w ∈ italic_W, the following are equivalent:

1) K(v)B(w)𝐾𝑣𝐵𝑤K(v)\cap B(w)\neq\emptysetitalic_K ( italic_v ) ∩ italic_B ( italic_w ) ≠ ∅;

2) K0(v)B(w)subscript𝐾0𝑣𝐵𝑤K_{0}(v)\cap B(w)\neq\emptysetitalic_K start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_v ) ∩ italic_B ( italic_w ) ≠ ∅.

Proof.

It follows from K=K0BK𝐾subscript𝐾0subscript𝐵𝐾K=K_{0}B_{K}italic_K = italic_K start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT that 1) and 2) are equivalent.

Q.E.D.

6. Review on K𝐾Kitalic_K-orbits in {\mathcal{B}}caligraphic_B, II

Keeping the notation as in §§\lx@sectionsign§3, we now review the canonical definitions of more structures on the set V𝑉Vitalic_V of K𝐾Kitalic_K-orbits in {\mathcal{B}}caligraphic_B, notably the cross action, the Springer map, root types, and reduced decompositions. All the results in this section, except Proposition 6.8 in §§\lx@sectionsign§6.8, can be found in [12, 13].

6.1. The canonical maximal torus Hcansubscript𝐻canH_{\rm can}italic_H start_POSTSUBSCRIPT roman_can end_POSTSUBSCRIPT and the W𝑊Witalic_W-action on Hcansubscript𝐻canH_{\rm can}italic_H start_POSTSUBSCRIPT roman_can end_POSTSUBSCRIPT

Recall from §§\lx@sectionsign§3.2 that 𝒞𝒞{\mathcal{C}}caligraphic_C is the variety of all pairs (B,H)𝐵𝐻(B,H)( italic_B , italic_H ), where B𝐵B\in{\mathcal{B}}italic_B ∈ caligraphic_B and HB𝐻𝐵H\subset Bitalic_H ⊂ italic_B is a maximal torus of G𝐺Gitalic_G. Let 𝒞~={(B,H,h)𝒞×G:(B,H)𝒞,hH}~𝒞conditional-set𝐵𝐻𝒞𝐺formulae-sequence𝐵𝐻𝒞𝐻\widetilde{{\mathcal{C}}}=\{(B,H,h)\in{\mathcal{C}}\times G:(B,H)\in{\mathcal{% C}},\,h\in H\}~ start_ARG caligraphic_C end_ARG = { ( italic_B , italic_H , italic_h ) ∈ caligraphic_C × italic_G : ( italic_B , italic_H ) ∈ caligraphic_C , italic_h ∈ italic_H }, and let G𝐺Gitalic_G act on 𝒞~~𝒞\widetilde{{\mathcal{C}}}~ start_ARG caligraphic_C end_ARG by

(B,H,h)g=(Bg,Hg,g-1hg),(B,H,h)𝒞~,gG.formulae-sequencesuperscript𝐵𝐻𝑔superscript𝐵𝑔superscript𝐻𝑔superscript𝑔1𝑔formulae-sequence𝐵𝐻~𝒞𝑔𝐺(B,H,h)^{g}=(B^{g},H^{g},g^{-1}hg),\hskip 14.454pt(B,H,h)\in\widetilde{{% \mathcal{C}}},\;g\in G.( italic_B , italic_H , italic_h ) start_POSTSUPERSCRIPT italic_g end_POSTSUPERSCRIPT = ( italic_B start_POSTSUPERSCRIPT italic_g end_POSTSUPERSCRIPT , italic_H start_POSTSUPERSCRIPT italic_g end_POSTSUPERSCRIPT , italic_g start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_h italic_g ) , ( italic_B , italic_H , italic_h ) ∈ ~ start_ARG caligraphic_C end_ARG , italic_g ∈ italic_G .

Let Hcan=𝒞~/Gsubscript𝐻can~𝒞𝐺H_{\rm can}=\widetilde{{\mathcal{C}}}/Gitalic_H start_POSTSUBSCRIPT roman_can end_POSTSUBSCRIPT = ~ start_ARG caligraphic_C end_ARG / italic_G be the set of G𝐺Gitalic_G-orbits in 𝒞~~𝒞\widetilde{{\mathcal{C}}}~ start_ARG caligraphic_C end_ARG, and let 𝒞~Hcan:(B,H,h)[B,H,h]:~𝒞subscript𝐻can𝐵𝐻𝐵𝐻\widetilde{{\mathcal{C}}}\to H_{\rm can}:(B,H,h)\to[B,H,h]~ start_ARG caligraphic_C end_ARG → italic_H start_POSTSUBSCRIPT roman_can end_POSTSUBSCRIPT : ( italic_B , italic_H , italic_h ) → [ italic_B , italic_H , italic_h ] be the canonical projection. Then for every (B,H)𝒞𝐵𝐻𝒞(B,H)\in{\mathcal{C}}( italic_B , italic_H ) ∈ caligraphic_C, one has the bijective map

TB,H:HHcan:h[B,H,h],hH.:subscript𝑇𝐵𝐻𝐻subscript𝐻can:formulae-sequence𝐵𝐻𝐻T_{B,H}:\;H\longrightarrow H_{\rm can}:\;\;h\longmapsto[B,H,h],\hskip 14.454% pth\in H.italic_T start_POSTSUBSCRIPT italic_B , italic_H end_POSTSUBSCRIPT : italic_H ⟶ italic_H start_POSTSUBSCRIPT roman_can end_POSTSUBSCRIPT : italic_h ⟼ [ italic_B , italic_H , italic_h ] , italic_h ∈ italic_H .

Since TB,H-1TB,H=TB,HB,H:HH:superscriptsubscript𝑇superscript𝐵superscript𝐻1subscript𝑇𝐵𝐻superscriptsubscript𝑇𝐵𝐻superscript𝐵superscript𝐻𝐻superscript𝐻T_{B^{\prime},H^{\prime}}^{-1}\circ T_{B,H}=T_{B,H}^{B^{\prime},H^{\prime}}:H% \to H^{\prime}italic_T start_POSTSUBSCRIPT italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_H start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ∘ italic_T start_POSTSUBSCRIPT italic_B , italic_H end_POSTSUBSCRIPT = italic_T start_POSTSUBSCRIPT italic_B , italic_H end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_H start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT : italic_H → italic_H start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT for any (B,H),(B,H)𝒞𝐵𝐻superscript𝐵superscript𝐻𝒞(B,H),(B^{\prime},H^{\prime})\in{\mathcal{C}}( italic_B , italic_H ) , ( italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_H start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ∈ caligraphic_C, where TB,HB,Hsuperscriptsubscript𝑇𝐵𝐻superscript𝐵superscript𝐻T_{B,H}^{B^{\prime},H^{\prime}}italic_T start_POSTSUBSCRIPT italic_B , italic_H end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_H start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT is the isomorphism of tori in (3.3), Hcansubscript𝐻canH_{\rm can}italic_H start_POSTSUBSCRIPT roman_can end_POSTSUBSCRIPT has a well-defined structure of a torus, such that TB,H:HHcan:subscript𝑇𝐵𝐻𝐻subscript𝐻canT_{B,H}:H\to H_{\rm can}italic_T start_POSTSUBSCRIPT italic_B , italic_H end_POSTSUBSCRIPT : italic_H → italic_H start_POSTSUBSCRIPT roman_can end_POSTSUBSCRIPT is an isomorphism of tori for every (B,H)𝒞𝐵𝐻𝒞(B,H)\in{\mathcal{C}}( italic_B , italic_H ) ∈ caligraphic_C. We will call Hcansubscript𝐻canH_{\rm can}italic_H start_POSTSUBSCRIPT roman_can end_POSTSUBSCRIPT the canonical maximal torus of G𝐺Gitalic_G.

The canonical Weyl group W𝑊Witalic_W acts on Hcansubscript𝐻canH_{\rm can}italic_H start_POSTSUBSCRIPT roman_can end_POSTSUBSCRIPT through the action of WHsubscript𝑊𝐻W_{H}italic_W start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT on H𝐻Hitalic_H via the identifications ηB,H-1:WWH:superscriptsubscript𝜂𝐵𝐻1𝑊subscript𝑊𝐻\eta_{B,H}^{-1}:W\to W_{H}italic_η start_POSTSUBSCRIPT italic_B , italic_H end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT : italic_W → italic_W start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT and TB,H-1:HcanH:superscriptsubscript𝑇𝐵𝐻1subscript𝐻can𝐻T_{B,H}^{-1}:H_{\rm can}\to Hitalic_T start_POSTSUBSCRIPT italic_B , italic_H end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT : italic_H start_POSTSUBSCRIPT roman_can end_POSTSUBSCRIPT → italic_H for any (B,H)𝒞𝐵𝐻𝒞(B,H)\in{\mathcal{C}}( italic_B , italic_H ) ∈ caligraphic_C, and the action is independent of the choice of (B,H)𝒞𝐵𝐻𝒞(B,H)\in{\mathcal{C}}( italic_B , italic_H ) ∈ caligraphic_C. If (B,H),(B,H)𝒞𝐵𝐻superscript𝐵𝐻𝒞(B,H),(B^{\prime},H)\in{\mathcal{C}}( italic_B , italic_H ) , ( italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_H ) ∈ caligraphic_C, then

(6.1) p(B,B)([B,H,h])=[B,H,h],hH.formulae-sequence𝑝superscript𝐵𝐵𝐵𝐻superscript𝐵𝐻𝐻p(B^{\prime},B)([B,H,h])=[B^{\prime},H,h],\hskip 14.454pth\in H.italic_p ( italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_B ) ( [ italic_B , italic_H , italic_h ] ) = [ italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_H , italic_h ] , italic_h ∈ italic_H .

One can also take (6.1) as the definition of the canonical action of W𝑊Witalic_W on Hcansubscript𝐻canH_{\rm can}italic_H start_POSTSUBSCRIPT roman_can end_POSTSUBSCRIPT.

6.2. The canonical root system of G𝐺Gitalic_G

Let Xcansubscript𝑋canX_{\rm can}italic_X start_POSTSUBSCRIPT roman_can end_POSTSUBSCRIPT be the character group of Hcansubscript𝐻canH_{\rm can}italic_H start_POSTSUBSCRIPT roman_can end_POSTSUBSCRIPT. For (B,H)𝒞𝐵𝐻𝒞(B,H)\in{\mathcal{C}}( italic_B , italic_H ) ∈ caligraphic_C, let XHsubscript𝑋𝐻X_{H}italic_X start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT be the character group of H𝐻Hitalic_H, and let RB,H:XHXcan:subscript𝑅𝐵𝐻subscript𝑋𝐻subscript𝑋canR_{B,H}:X_{H}\to X_{\rm can}italic_R start_POSTSUBSCRIPT italic_B , italic_H end_POSTSUBSCRIPT : italic_X start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT → italic_X start_POSTSUBSCRIPT roman_can end_POSTSUBSCRIPT be the isomorphism induced by TB,H-1:HcanH:superscriptsubscript𝑇𝐵𝐻1subscript𝐻can𝐻T_{B,H}^{-1}:H_{\rm can}\to Hitalic_T start_POSTSUBSCRIPT italic_B , italic_H end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT : italic_H start_POSTSUBSCRIPT roman_can end_POSTSUBSCRIPT → italic_H. The canonical sets of roots, positive roots, and simple roots of G𝐺Gitalic_G, are the subsets of Xcansubscript𝑋canX_{\rm can}italic_X start_POSTSUBSCRIPT roman_can end_POSTSUBSCRIPT, respectively defined by

Δ=RB,H(ΔH),Δ+=RB,H(ΔB,H+),Γ=RB,H(ΓB,H),formulae-sequenceΔsubscript𝑅𝐵𝐻subscriptΔ𝐻formulae-sequencesuperscriptΔsubscript𝑅𝐵𝐻subscriptsuperscriptΔ𝐵𝐻Γsubscript𝑅𝐵𝐻subscriptΓ𝐵𝐻\Delta=R_{B,H}(\Delta_{H}),\hskip 14.454pt\Delta^{+}=R_{B,H}(\Delta^{+}_{B,H})% ,\hskip 14.454pt\Gamma=R_{B,H}(\Gamma_{B,H}),roman_Δ = italic_R start_POSTSUBSCRIPT italic_B , italic_H end_POSTSUBSCRIPT ( roman_Δ start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT ) , roman_Δ start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT = italic_R start_POSTSUBSCRIPT italic_B , italic_H end_POSTSUBSCRIPT ( roman_Δ start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_B , italic_H end_POSTSUBSCRIPT ) , roman_Γ = italic_R start_POSTSUBSCRIPT italic_B , italic_H end_POSTSUBSCRIPT ( roman_Γ start_POSTSUBSCRIPT italic_B , italic_H end_POSTSUBSCRIPT ) ,

where (B,H)𝒞𝐵𝐻𝒞(B,H)\in{\mathcal{C}}( italic_B , italic_H ) ∈ caligraphic_C, and ΔHΔB,H+ΓB,Hsuperset-ofsubscriptΔ𝐻subscriptsuperscriptΔ𝐵𝐻superset-ofsubscriptΓ𝐵𝐻\Delta_{H}\supset\Delta^{+}_{B,H}\supset\Gamma_{B,H}roman_Δ start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT ⊃ roman_Δ start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_B , italic_H end_POSTSUBSCRIPT ⊃ roman_Γ start_POSTSUBSCRIPT italic_B , italic_H end_POSTSUBSCRIPT are the subsets of the character group XHsubscript𝑋𝐻X_{H}italic_X start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT of H𝐻Hitalic_H consisting, respectively, of the roots of (G,H)𝐺𝐻(G,H)( italic_G , italic_H ), the positive roots, and the simple roots determined by B𝐵Bitalic_B. It is easy to check that the sets ΓΔ+ΔXcanΓsuperscriptΔΔsubscript𝑋can\Gamma\subset\Delta^{+}\subset\Delta\subset X_{\rm can}roman_Γ ⊂ roman_Δ start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ⊂ roman_Δ ⊂ italic_X start_POSTSUBSCRIPT roman_can end_POSTSUBSCRIPT are independent of the choice of (B,H)𝒞𝐵𝐻𝒞(B,H)\in{\mathcal{C}}( italic_B , italic_H ) ∈ caligraphic_C.

For αΔ𝛼Δ\alpha\in\Deltaitalic_α ∈ roman_Δ, choose any (B,H)𝒞𝐵𝐻𝒞(B,H)\in{\mathcal{C}}( italic_B , italic_H ) ∈ caligraphic_C, let αH=RB,H-1(α)ΔHsubscript𝛼𝐻superscriptsubscript𝑅𝐵𝐻1𝛼subscriptΔ𝐻\alpha_{H}=R_{B,H}^{-1}(\alpha)\in\Delta_{H}italic_α start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT = italic_R start_POSTSUBSCRIPT italic_B , italic_H end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_α ) ∈ roman_Δ start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT, let sαHWHsubscript𝑠subscript𝛼𝐻subscript𝑊𝐻s_{\alpha_{H}}\in W_{H}italic_s start_POSTSUBSCRIPT italic_α start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∈ italic_W start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT be the reflection defined by αHsubscript𝛼𝐻\alpha_{H}italic_α start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT, and let sα=ηB,H(sαH)Wsubscript𝑠𝛼subscript𝜂𝐵𝐻subscript𝑠subscript𝛼𝐻𝑊s_{\alpha}=\eta_{B,H}(s_{\alpha_{H}})\in Witalic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT = italic_η start_POSTSUBSCRIPT italic_B , italic_H end_POSTSUBSCRIPT ( italic_s start_POSTSUBSCRIPT italic_α start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) ∈ italic_W. Then sαsubscript𝑠𝛼s_{\alpha}italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT is independent of the choice of (B,H)𝒞𝐵𝐻𝒞(B,H)\in{\mathcal{C}}( italic_B , italic_H ) ∈ caligraphic_C. Moreover, S={sα:αΓ}𝑆conditional-setsubscript𝑠𝛼𝛼ΓS=\{s_{\alpha}:\alpha\in\Gamma\}italic_S = { italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT : italic_α ∈ roman_Γ }.

6.3. The automorphism θ𝜃\thetaitalic_θ on Hcansubscript𝐻canH_{\rm can}italic_H start_POSTSUBSCRIPT roman_can end_POSTSUBSCRIPT, ΓΓ\Gammaroman_Γ, and W𝑊Witalic_W

In this subsection, let θ𝜃\thetaitalic_θ be any automorphism of G𝐺Gitalic_G, not necessarily of order 2222. Then one has the well-defined map

θ:HcanHcan:[B,H,h][θ(B),θ(H),θ(h)],(B,H,h)𝒞~.:𝜃subscript𝐻cansubscript𝐻can:formulae-sequence𝐵𝐻𝜃𝐵𝜃𝐻𝜃𝐵𝐻~𝒞\theta:\;H_{\rm can}\longrightarrow H_{\rm can}:\;\;[B,H,h]\longmapsto[\theta(% B),\theta(H),\theta(h)],\hskip 14.454pt(B,H,h)\in\widetilde{{\mathcal{C}}}.italic_θ : italic_H start_POSTSUBSCRIPT roman_can end_POSTSUBSCRIPT ⟶ italic_H start_POSTSUBSCRIPT roman_can end_POSTSUBSCRIPT : [ italic_B , italic_H , italic_h ] ⟼ [ italic_θ ( italic_B ) , italic_θ ( italic_H ) , italic_θ ( italic_h ) ] , ( italic_B , italic_H , italic_h ) ∈ ~ start_ARG caligraphic_C end_ARG .

If (B,H)𝒞𝐵𝐻𝒞(B,H)\in{\mathcal{C}}( italic_B , italic_H ) ∈ caligraphic_C, then, by definition, θ=Tθ(B),θ(H)θ|HTB,H-1𝜃evaluated-atsubscript𝑇𝜃𝐵𝜃𝐻𝜃𝐻superscriptsubscript𝑇𝐵𝐻1\theta=T_{\theta(B),\theta(H)}\circ\theta|_{H}\circ T_{B,H}^{-1}italic_θ = italic_T start_POSTSUBSCRIPT italic_θ ( italic_B ) , italic_θ ( italic_H ) end_POSTSUBSCRIPT ∘ italic_θ | start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT ∘ italic_T start_POSTSUBSCRIPT italic_B , italic_H end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT, where θ|H:Hθ(H):evaluated-at𝜃𝐻𝐻𝜃𝐻\theta|_{H}:H\to\theta(H)italic_θ | start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT : italic_H → italic_θ ( italic_H ). Thus θ𝜃\thetaitalic_θ is an automorphism of Hcansubscript𝐻canH_{\rm can}italic_H start_POSTSUBSCRIPT roman_can end_POSTSUBSCRIPT.

It is clear from the definition that if θ𝜃\thetaitalic_θ is inner, i.e., if there exists g1Gsubscript𝑔1𝐺g_{1}\in Gitalic_g start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∈ italic_G such that θ(g)=g1gg1-1𝜃𝑔subscript𝑔1𝑔superscriptsubscript𝑔11\theta(g)=g_{1}gg_{1}^{-1}italic_θ ( italic_g ) = italic_g start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_g italic_g start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT for gG𝑔𝐺g\in Gitalic_g ∈ italic_G, then θ𝜃\thetaitalic_θ induces the identity automorphism of Hcansubscript𝐻canH_{\rm can}italic_H start_POSTSUBSCRIPT roman_can end_POSTSUBSCRIPT.

We will use the same letter to denote the induced action of θ𝜃\thetaitalic_θ on the set ΔΔ\Deltaroman_Δ of canonical roots. It follows from the definitions that θ(Δ+)=Δ+𝜃superscriptΔsuperscriptΔ\theta(\Delta^{+})=\Delta^{+}italic_θ ( roman_Δ start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ) = roman_Δ start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT and θ(Γ)=Γ𝜃ΓΓ\theta(\Gamma)=\Gammaitalic_θ ( roman_Γ ) = roman_Γ.

The following map, again denoted by θ𝜃\thetaitalic_θ, is easily seen to be well-defined:

θ:WW:p(B1,B2)p(θ(B1),θ(B2)),B1,B2,:𝜃𝑊𝑊:formulae-sequence𝑝subscript𝐵1subscript𝐵2𝑝𝜃subscript𝐵1𝜃subscript𝐵2subscript𝐵1subscript𝐵2\theta:\;W\longrightarrow W:\;\;\;p(B_{1},B_{2})\longmapsto p(\theta(B_{1}),% \theta(B_{2})),\hskip 14.454ptB_{1},B_{2}\in{\mathcal{B}},italic_θ : italic_W ⟶ italic_W : italic_p ( italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_B start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ⟼ italic_p ( italic_θ ( italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) , italic_θ ( italic_B start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ) , italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_B start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∈ caligraphic_B ,

Choose any (B,H)𝒞𝐵𝐻𝒞(B,H)\in{\mathcal{C}}( italic_B , italic_H ) ∈ caligraphic_C, and let θWH:WHWθ(H):subscript𝜃subscript𝑊𝐻subscript𝑊𝐻subscript𝑊𝜃𝐻\theta_{W_{H}}:W_{H}\to W_{\theta(H)}italic_θ start_POSTSUBSCRIPT italic_W start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT end_POSTSUBSCRIPT : italic_W start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT → italic_W start_POSTSUBSCRIPT italic_θ ( italic_H ) end_POSTSUBSCRIPT be the group isomorphism induced by θ|NG(H):NG(H)NG(θ(H)):evaluated-at𝜃subscript𝑁𝐺𝐻subscript𝑁𝐺𝐻subscript𝑁𝐺𝜃𝐻\theta|_{N_{G}(H)}:N_{G}(H)\to N_{G}(\theta(H))italic_θ | start_POSTSUBSCRIPT italic_N start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_H ) end_POSTSUBSCRIPT : italic_N start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_H ) → italic_N start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_θ ( italic_H ) ). It follows from definitions that

θ=ηθ(B),θ(H)θWHηB,H-1:WW.:𝜃subscript𝜂𝜃𝐵𝜃𝐻subscript𝜃subscript𝑊𝐻superscriptsubscript𝜂𝐵𝐻1𝑊𝑊\theta=\eta_{\theta(B),\theta(H)}\circ\theta_{W_{H}}\circ\eta_{B,H}^{-1}:\;\;W% \longrightarrow W.italic_θ = italic_η start_POSTSUBSCRIPT italic_θ ( italic_B ) , italic_θ ( italic_H ) end_POSTSUBSCRIPT ∘ italic_θ start_POSTSUBSCRIPT italic_W start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∘ italic_η start_POSTSUBSCRIPT italic_B , italic_H end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT : italic_W ⟶ italic_W .

Thus θ𝜃\thetaitalic_θ is an automorphism of W𝑊Witalic_W. It also follows that θ(sα)=sθ(α)𝜃subscript𝑠𝛼subscript𝑠𝜃𝛼\theta(s_{\alpha})=s_{\theta(\alpha)}italic_θ ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ) = italic_s start_POSTSUBSCRIPT italic_θ ( italic_α ) end_POSTSUBSCRIPT for all αΔ𝛼Δ\alpha\in\Deltaitalic_α ∈ roman_Δ.

6.4. The identification γ:𝒞θ/KV:𝛾subscript𝒞𝜃𝐾𝑉\gamma:{\mathcal{C}}_{\theta}/K\to Vitalic_γ : caligraphic_C start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT / italic_K → italic_V

Let 𝒯θsuperscript𝒯𝜃{\mathcal{T}}^{\theta}caligraphic_T start_POSTSUPERSCRIPT italic_θ end_POSTSUPERSCRIPT be the set of all θ𝜃\thetaitalic_θ-stable maximal tori in G𝐺Gitalic_G. The following Lemma 6.1 is proved in [13, Proposition 1.2.1] and [16, Corollary 4.4]. See also [12, 1.4(b)].

Lemma 6.1.

Every B𝐵B\in{\mathcal{B}}italic_B ∈ caligraphic_B contains some H𝒯θ𝐻superscript𝒯𝜃H\in{\mathcal{T}}^{\theta}italic_H ∈ caligraphic_T start_POSTSUPERSCRIPT italic_θ end_POSTSUPERSCRIPT, and if H1,H2𝒯θsubscript𝐻1subscript𝐻2superscript𝒯𝜃H_{1},H_{2}\in{\mathcal{T}}^{\theta}italic_H start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_H start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∈ caligraphic_T start_POSTSUPERSCRIPT italic_θ end_POSTSUPERSCRIPT are both contained in B𝐵Bitalic_B, then H1=k-1H2ksubscript𝐻1superscript𝑘1subscript𝐻2𝑘H_{1}=k^{-1}H_{2}kitalic_H start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_k start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_H start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_k for some kBK𝑘𝐵𝐾k\in B\cap Kitalic_k ∈ italic_B ∩ italic_K.

Let 𝒞θ={(B,H)𝒞:H𝒯θ}subscript𝒞𝜃conditional-set𝐵𝐻𝒞𝐻superscript𝒯𝜃{\mathcal{C}}_{\theta}=\{(B,H)\in{\mathcal{C}}:H\in{\mathcal{T}}^{\theta}\}caligraphic_C start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT = { ( italic_B , italic_H ) ∈ caligraphic_C : italic_H ∈ caligraphic_T start_POSTSUPERSCRIPT italic_θ end_POSTSUPERSCRIPT } and let K𝐾Kitalic_K act on 𝒞θsubscript𝒞𝜃{\mathcal{C}}_{\theta}caligraphic_C start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT by (3.1). It follows from Lemma 6.1 that the well-defined map

γ:𝒞θ/KV=/K:𝐾-orbit of(B,H)in𝒞𝐾-orbit ofBin,:𝛾subscript𝒞𝜃𝐾𝑉𝐾:𝐾-orbit of𝐵𝐻in𝒞𝐾-orbit of𝐵in\gamma:\;{\mathcal{C}}_{\theta}/K\longrightarrow V={\mathcal{B}}/K:\;\mbox{{% \it K}-orbit of}\;(B,H)\;\mbox{in}\;{\mathcal{C}}\longmapsto\mbox{{\it K}-% orbit of}\;B\;\mbox{in}\;{\mathcal{B}},italic_γ : caligraphic_C start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT / italic_K ⟶ italic_V = caligraphic_B / italic_K : italic_K -orbit of ( italic_B , italic_H ) in caligraphic_C ⟼ italic_K -orbit of italic_B in caligraphic_B ,

is a bijection ([13, Proposition 1.2.1] and [16, Corollary 4.4]).

6.5. The cross action of W𝑊Witalic_W on V𝑉Vitalic_V

We follow [12, §§\lx@sectionsign§2] and [13, §§\lx@sectionsign§1.7]. See also [4]. For any (B,H)𝒞𝐵𝐻𝒞(B,H)\in{\mathcal{C}}( italic_B , italic_H ) ∈ caligraphic_C and wW𝑤𝑊w\in Witalic_w ∈ italic_W, there is a unique (B,H)𝒞superscript𝐵𝐻𝒞(B^{\prime},H)\in{\mathcal{C}}( italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_H ) ∈ caligraphic_C such that w=p(B,B)𝑤𝑝superscript𝐵𝐵w=p(B^{\prime},B)italic_w = italic_p ( italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_B ). Define

(6.2) W×𝒞𝒞:w(B,H)=(B,H),(B,H),(B,H)𝒞,w=p(B,B).:𝑊𝒞𝒞formulae-sequence𝑤𝐵𝐻superscript𝐵𝐻𝐵𝐻formulae-sequencesuperscript𝐵𝐻𝒞𝑤𝑝superscript𝐵𝐵W\times{\mathcal{C}}\longrightarrow{\mathcal{C}}:\;\;w\cdot(B,H)=(B^{\prime},H% ),\hskip 14.454pt(B,H),(B^{\prime},H)\in{\mathcal{C}},\,w=p(B^{\prime},B).italic_W × caligraphic_C ⟶ caligraphic_C : italic_w ⋅ ( italic_B , italic_H ) = ( italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_H ) , ( italic_B , italic_H ) , ( italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_H ) ∈ caligraphic_C , italic_w = italic_p ( italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_B ) .

If we fix (B,H)𝒞𝐵𝐻𝒞(B,H)\in{\mathcal{C}}( italic_B , italic_H ) ∈ caligraphic_C, then under the identifications ηB,H:WHW:subscript𝜂𝐵𝐻subscript𝑊𝐻𝑊\eta_{B,H}:W_{H}\to Witalic_η start_POSTSUBSCRIPT italic_B , italic_H end_POSTSUBSCRIPT : italic_W start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT → italic_W and CB,H:H\G𝒞:Hg(Bg,Hg):subscript𝐶𝐵𝐻\𝐻𝐺𝒞:maps-to𝐻𝑔superscript𝐵𝑔superscript𝐻𝑔C_{B,H}:H\backslash G\to{\mathcal{C}}:Hg\mapsto(B^{g},H^{g})italic_C start_POSTSUBSCRIPT italic_B , italic_H end_POSTSUBSCRIPT : italic_H \ italic_G → caligraphic_C : italic_H italic_g ↦ ( italic_B start_POSTSUPERSCRIPT italic_g end_POSTSUPERSCRIPT , italic_H start_POSTSUPERSCRIPT italic_g end_POSTSUPERSCRIPT ) for gG𝑔𝐺g\in Gitalic_g ∈ italic_G, the map in (6.2) becomes

WH×H\GH\G:(nH,Hg)Hng,nNG(H),gG,:\subscript𝑊𝐻𝐻𝐺\𝐻𝐺formulae-sequence𝑛𝐻𝐻𝑔𝐻𝑛𝑔formulae-sequence𝑛subscript𝑁𝐺𝐻𝑔𝐺W_{H}\times H\backslash G\longrightarrow H\backslash G:\;\;\;(nH,Hg)% \longmapsto Hng,\hskip 14.454ptn\in N_{G}(H),\,g\in G,italic_W start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT × italic_H \ italic_G ⟶ italic_H \ italic_G : ( italic_n italic_H , italic_H italic_g ) ⟼ italic_H italic_n italic_g , italic_n ∈ italic_N start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_H ) , italic_g ∈ italic_G ,

which is a left action of WHsubscript𝑊𝐻W_{H}italic_W start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT on H\G\𝐻𝐺H\backslash Gitalic_H \ italic_G. Thus (6.2) is indeed a left action of W𝑊Witalic_W on 𝒞𝒞{\mathcal{C}}caligraphic_C. Moreover, the W𝑊Witalic_W-action commutes with the right action of G𝐺Gitalic_G on 𝒞𝒞{\mathcal{C}}caligraphic_C given in (3.1).

By (6.2), 𝒞θ𝒞subscript𝒞𝜃𝒞{\mathcal{C}}_{\theta}\subset{\mathcal{C}}caligraphic_C start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT ⊂ caligraphic_C is W𝑊Witalic_W-invariant. Thus one has a well-defined action of W𝑊Witalic_W on 𝒞θ/Ksubscript𝒞𝜃𝐾{\mathcal{C}}_{\theta}/Kcaligraphic_C start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT / italic_K. Identifying V𝑉Vitalic_V with 𝒞θ/Ksubscript𝒞𝜃𝐾{\mathcal{C}}_{\theta}/Kcaligraphic_C start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT / italic_K via γ𝛾\gammaitalic_γ, one gets a left action of W𝑊Witalic_W on V𝑉Vitalic_V, which is called the cross action of W𝑊Witalic_W on V𝑉Vitalic_V and will be denoted by

W×VV:(w,v)wv,wW,vV.:𝑊𝑉𝑉formulae-sequence𝑤𝑣𝑤𝑣formulae-sequence𝑤𝑊𝑣𝑉W\times V\longrightarrow V:\;\;\;(w,v)\longmapsto w\!\cdot\!v,\hskip 14.454ptw% \in W,\,v\in V.italic_W × italic_V ⟶ italic_V : ( italic_w , italic_v ) ⟼ italic_w ⋅ italic_v , italic_w ∈ italic_W , italic_v ∈ italic_V .

6.6. The Springer map ϕ:VW:italic-ϕ𝑉𝑊\phi:V\to Witalic_ϕ : italic_V → italic_W

The Springer map ϕ:VW:italic-ϕ𝑉𝑊\phi:V\to Witalic_ϕ : italic_V → italic_W, introduced by Springer in [16], is defined (see [12, Remark 1.8] and [13, Proposition 1.7.1]) by

(6.3) ϕ(v)=p(B,θ(B)),vV,BK(v).formulae-sequenceitalic-ϕ𝑣𝑝𝐵𝜃𝐵formulae-sequence𝑣𝑉𝐵𝐾𝑣\phi(v)=p(B,\theta(B)),\hskip 14.454ptv\in V,\;B\in K(v).italic_ϕ ( italic_v ) = italic_p ( italic_B , italic_θ ( italic_B ) ) , italic_v ∈ italic_V , italic_B ∈ italic_K ( italic_v ) .

The element ϕ(v)Witalic-ϕ𝑣𝑊\phi(v)\in Witalic_ϕ ( italic_v ) ∈ italic_W for vV𝑣𝑉v\in Vitalic_v ∈ italic_V is an important invariant of the K𝐾Kitalic_K-orbit K(v)𝐾𝑣K(v)\subset{\mathcal{B}}italic_K ( italic_v ) ⊂ caligraphic_B. Recall that V0subscript𝑉0V_{0}italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT consists of all vV𝑣𝑉v\in Vitalic_v ∈ italic_V such that K(v)𝐾𝑣K(v)italic_K ( italic_v ) is closed in {\mathcal{B}}caligraphic_B. Let 1111 be the identity element of W𝑊Witalic_W. The following Proposition 6.2 is from [13, Proposition 1.4.2] and [12, Proposition 2.5].

Proposition 6.2.

1) For vV𝑣𝑉v\in Vitalic_v ∈ italic_V, vV0𝑣subscript𝑉0v\in V_{0}italic_v ∈ italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT if and only if ϕ(v)=1italic-ϕ𝑣1\phi(v)=1italic_ϕ ( italic_v ) = 1.

2) If v,vV𝑣superscript𝑣normal-′𝑉v,v^{\prime}\in Vitalic_v , italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_V are such that ϕ(v)=ϕ(v)italic-ϕ𝑣italic-ϕsuperscript𝑣normal-′\phi(v)=\phi(v^{\prime})italic_ϕ ( italic_v ) = italic_ϕ ( italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ), then v𝑣vitalic_v and vsuperscript𝑣normal-′v^{\prime}italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT are in the same W𝑊Witalic_W-orbit.

6.7. Using standard pairs

Let (B,H)𝒞𝐵𝐻𝒞(B,H)\in{\mathcal{C}}( italic_B , italic_H ) ∈ caligraphic_C be a standard pair, i.e., θ(B)=B𝜃𝐵𝐵\theta(B)=Bitalic_θ ( italic_B ) = italic_B and θ(H)=H𝜃𝐻𝐻\theta(H)=Hitalic_θ ( italic_H ) = italic_H. Then for any gG𝑔𝐺g\in Gitalic_g ∈ italic_G, (Bg,Hg)𝒞θsuperscript𝐵𝑔superscript𝐻𝑔subscript𝒞𝜃(B^{g},H^{g})\in{\mathcal{C}}_{\theta}( italic_B start_POSTSUPERSCRIPT italic_g end_POSTSUPERSCRIPT , italic_H start_POSTSUPERSCRIPT italic_g end_POSTSUPERSCRIPT ) ∈ caligraphic_C start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT if and only if gθ(g)-1NG(H)𝑔𝜃superscript𝑔1subscript𝑁𝐺𝐻g\theta(g)^{-1}\in N_{G}(H)italic_g italic_θ ( italic_g ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ∈ italic_N start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_H ), and for such a gG𝑔𝐺g\in Gitalic_g ∈ italic_G,

p(Bg,θ(Bg))=p(Bg,Bθ(g))=p(Bgθ(g)-1,B)=ηB,H(gθ(g)-1H)W.𝑝superscript𝐵𝑔𝜃superscript𝐵𝑔𝑝superscript𝐵𝑔superscript𝐵𝜃𝑔𝑝superscript𝐵𝑔𝜃superscript𝑔1𝐵subscript𝜂𝐵𝐻𝑔𝜃superscript𝑔1𝐻𝑊p(B^{g},\,\theta(B^{g}))=p(B^{g},\,B^{\theta(g)})=p(B^{g\theta(g)^{-1}},B)=% \eta_{B,H}(g\theta(g)^{-1}H)\in W.italic_p ( italic_B start_POSTSUPERSCRIPT italic_g end_POSTSUPERSCRIPT , italic_θ ( italic_B start_POSTSUPERSCRIPT italic_g end_POSTSUPERSCRIPT ) ) = italic_p ( italic_B start_POSTSUPERSCRIPT italic_g end_POSTSUPERSCRIPT , italic_B start_POSTSUPERSCRIPT italic_θ ( italic_g ) end_POSTSUPERSCRIPT ) = italic_p ( italic_B start_POSTSUPERSCRIPT italic_g italic_θ ( italic_g ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT , italic_B ) = italic_η start_POSTSUBSCRIPT italic_B , italic_H end_POSTSUBSCRIPT ( italic_g italic_θ ( italic_g ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_H ) ∈ italic_W .

Letting 𝒱H={gG:gθ(g)-1NG(H)}subscript𝒱𝐻conditional-set𝑔𝐺𝑔𝜃superscript𝑔1subscript𝑁𝐺𝐻{\mathcal{V}}_{H}=\{g\in G:\,g\theta(g)^{-1}\in N_{G}(H)\}caligraphic_V start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT = { italic_g ∈ italic_G : italic_g italic_θ ( italic_g ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ∈ italic_N start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_H ) }, one thus has the identification

(6.4) H\𝒱H/KV:HgKthe𝐾-orbit inthroughBg.:\𝐻subscript𝒱𝐻𝐾𝑉𝐻𝑔𝐾the𝐾-orbit inthroughsuperscript𝐵𝑔H\backslash{\mathcal{V}}_{H}/K\longrightarrow V:\;\;\;HgK\longmapsto\mbox{the}% \;\mbox{{\it K}-orbit in}\;{\mathcal{B}}\;\mbox{through}\;B^{g}.italic_H \ caligraphic_V start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT / italic_K ⟶ italic_V : italic_H italic_g italic_K ⟼ the italic_K -orbit in caligraphic_B through italic_B start_POSTSUPERSCRIPT italic_g end_POSTSUPERSCRIPT .

Under the identification of V𝑉Vitalic_V with H\𝒱H/K\𝐻subscript𝒱𝐻𝐾H\backslash{\mathcal{V}}_{H}/Kitalic_H \ caligraphic_V start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT / italic_K in (6.4) and that of W𝑊Witalic_W with WHsubscript𝑊𝐻W_{H}italic_W start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT by ηB,Hsubscript𝜂𝐵𝐻\eta_{B,H}italic_η start_POSTSUBSCRIPT italic_B , italic_H end_POSTSUBSCRIPT, the Springer map ϕ:VW:italic-ϕ𝑉𝑊\phi:V\to Witalic_ϕ : italic_V → italic_W becomes ϕ(HgK)=gθ(g)-1Hitalic-ϕ𝐻𝑔𝐾𝑔𝜃superscript𝑔1𝐻\phi(HgK)=g\theta(g)^{-1}Hitalic_ϕ ( italic_H italic_g italic_K ) = italic_g italic_θ ( italic_g ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_H for g𝒱H𝑔subscript𝒱𝐻g\in{\mathcal{V}}_{H}italic_g ∈ caligraphic_V start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT, and the action of W𝑊Witalic_W on V𝑉Vitalic_V becomes (nH)(HgK)=HngK𝑛𝐻𝐻𝑔𝐾𝐻𝑛𝑔𝐾(nH)\!\cdot\!(HgK)=HngK( italic_n italic_H ) ⋅ ( italic_H italic_g italic_K ) = italic_H italic_n italic_g italic_K for nNG(H)𝑛subscript𝑁𝐺𝐻n\in N_{G}(H)italic_n ∈ italic_N start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_H ) and g𝒱H𝑔subscript𝒱𝐻g\in{\mathcal{V}}_{H}italic_g ∈ caligraphic_V start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT.

In [12, 13, 16], most of the structures on V𝑉Vitalic_V are introduced and their properties proved by using standard pairs. For example, the following Lemma 6.3 immediately follows from the the identification (6.4) (see also [12, Lemma 2.1]).

Lemma 6.3.

One has ϕ(wv)=wϕ(v)θ(w)-1italic-ϕnormal-⋅𝑤𝑣𝑤italic-ϕ𝑣𝜃superscript𝑤1\phi(w\!\cdot\!v)=w\phi(v)\theta(w)^{-1}italic_ϕ ( italic_w ⋅ italic_v ) = italic_w italic_ϕ ( italic_v ) italic_θ ( italic_w ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT for any wW𝑤𝑊w\in Witalic_w ∈ italic_W and vV𝑣𝑉v\in Vitalic_v ∈ italic_V.

6.8. The closed K𝐾Kitalic_K-orbits in {\mathcal{B}}caligraphic_B

Let Wθ={wW:θ(w)=w}superscript𝑊𝜃conditional-set𝑤𝑊𝜃𝑤𝑤W^{\theta}=\{w\in W:\theta(w)=w\}italic_W start_POSTSUPERSCRIPT italic_θ end_POSTSUPERSCRIPT = { italic_w ∈ italic_W : italic_θ ( italic_w ) = italic_w }. By Lemma 6.3,

(6.5) {wW:wv0=v0}={wWθ:wv0=v0},v0,v0V0.formulae-sequenceconditional-set𝑤𝑊𝑤subscript𝑣0superscriptsubscript𝑣0conditional-set𝑤superscript𝑊𝜃𝑤subscript𝑣0superscriptsubscript𝑣0for-allsubscript𝑣0superscriptsubscript𝑣0subscript𝑉0\{w\in W:\;w\!\cdot\!v_{0}=v_{0}^{\prime}\}=\{w\in W^{\theta}:\;w\!\cdot\!v_{0% }=v_{0}^{\prime}\},\hskip 14.454pt\forall\,v_{0},v_{0}^{\prime}\in V_{0}.{ italic_w ∈ italic_W : italic_w ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT } = { italic_w ∈ italic_W start_POSTSUPERSCRIPT italic_θ end_POSTSUPERSCRIPT : italic_w ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT } , ∀ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT .

By Proposition 6.2, Wθsuperscript𝑊𝜃W^{\theta}italic_W start_POSTSUPERSCRIPT italic_θ end_POSTSUPERSCRIPT acts transitively on V0subscript𝑉0V_{0}italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT.

For the rest of this subsection, we assume that K𝐾Kitalic_K is connected. We will relate the sets in (6.5) for v0,v0V0subscript𝑣0superscriptsubscript𝑣0subscript𝑉0v_{0},v_{0}^{\prime}\in V_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT with the canonical Weyl group WKsubscript𝑊𝐾W_{K}italic_W start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT of K𝐾Kitalic_K. Proposition 6.8 will be used in §§\lx@sectionsign§9.1, where we determine Yv0(v0)subscript𝑌subscript𝑣0superscriptsubscript𝑣0Y_{v_{0}}(v_{0}^{\prime})italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) for every v0,v0V0subscript𝑣0superscriptsubscript𝑣0subscript𝑉0v_{0},v_{0}^{\prime}\in V_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT.

Lemma 6.4.

Let v0,v0V0subscript𝑣0superscriptsubscript𝑣0normal-′subscript𝑉0v_{0},v_{0}^{\prime}\in V_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT. Then for any BK(v0)𝐵𝐾subscript𝑣0B\in K(v_{0})italic_B ∈ italic_K ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) and BK(v0)superscript𝐵normal-′𝐾superscriptsubscript𝑣0normal-′B^{\prime}\in K(v_{0}^{\prime})italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_K ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ), there exists H𝒯θ𝐻superscript𝒯𝜃H\in{\mathcal{T}}^{\theta}italic_H ∈ caligraphic_T start_POSTSUPERSCRIPT italic_θ end_POSTSUPERSCRIPT such that HBB𝐻𝐵superscript𝐵normal-′H\subset B\cap B^{\prime}italic_H ⊂ italic_B ∩ italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT.

Proof.

Since K(v0)𝐾subscript𝑣0K(v_{0})italic_K ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) and K(v0)𝐾superscriptsubscript𝑣0K(v_{0}^{\prime})italic_K ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) are closed, it follows from [13, Theorem 1.4.3] (see also [16, Corollary 6.6]) that B𝐵Bitalic_B and Bsuperscript𝐵B^{\prime}italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT are θ𝜃\thetaitalic_θ-stable. Thus, BB𝐵superscript𝐵B\cap B^{\prime}italic_B ∩ italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT is θ𝜃\thetaitalic_θ-stable and contains a maximal torus of G𝐺Gitalic_G. Hence, by [18, Theorem 7.5], BB𝐵superscript𝐵B\cap B^{\prime}italic_B ∩ italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT contains a θ𝜃\thetaitalic_θ-stable maximal torus of G𝐺Gitalic_G.

Q.E.D.

Let v0,v0V0subscript𝑣0superscriptsubscript𝑣0subscript𝑉0v_{0},v_{0}^{\prime}\in V_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and consider the restriction p:K(v0)×K(v0)W:𝑝𝐾superscriptsubscript𝑣0𝐾subscript𝑣0𝑊p:K(v_{0}^{\prime})\times K(v_{0})\rightarrow Witalic_p : italic_K ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) × italic_K ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) → italic_W. By definition,

p(K(v0)×K(v0))={wW:w=p(B,B)for someBK(v0),BK(v0)}.𝑝𝐾superscriptsubscript𝑣0𝐾subscript𝑣0conditional-set𝑤𝑊formulae-sequence𝑤𝑝superscript𝐵𝐵for somesuperscript𝐵𝐾superscriptsubscript𝑣0𝐵𝐾subscript𝑣0p(K(v_{0}^{\prime})\times K(v_{0}))=\{w\in W:w=p(B^{\prime},B)\,\mbox{for some% }\;B^{\prime}\in K(v_{0}^{\prime}),\,B\in K(v_{0})\}.italic_p ( italic_K ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) × italic_K ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) ) = { italic_w ∈ italic_W : italic_w = italic_p ( italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_B ) for some italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_K ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) , italic_B ∈ italic_K ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) } .
Lemma 6.5.

For any v0,v0V0subscript𝑣0superscriptsubscript𝑣0normal-′subscript𝑉0v_{0},v_{0}^{\prime}\in V_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, one has

p(K(v0)×K(v0))={wW:wv0=v0}.𝑝𝐾superscriptsubscript𝑣0𝐾subscript𝑣0conditional-set𝑤𝑊𝑤subscript𝑣0superscriptsubscript𝑣0p(K(v_{0}^{\prime})\times K(v_{0}))=\{w\in W:w\!\cdot\!v_{0}=v_{0}^{\prime}\}.italic_p ( italic_K ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) × italic_K ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) ) = { italic_w ∈ italic_W : italic_w ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT } .
Proof.

By the definition of the W𝑊Witalic_W-action, {wW:wv0=v0}p(K(v0)×K(v0))conditional-set𝑤𝑊𝑤subscript𝑣0superscriptsubscript𝑣0𝑝𝐾superscriptsubscript𝑣0𝐾subscript𝑣0\{w\in W:w\!\cdot\!v_{0}=v_{0}^{\prime}\}\subset p(K(v_{0}^{\prime})\times K(v% _{0})){ italic_w ∈ italic_W : italic_w ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT } ⊂ italic_p ( italic_K ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) × italic_K ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) ). Suppose that w=p(B,B)𝑤𝑝superscript𝐵𝐵w=p(B^{\prime},B)italic_w = italic_p ( italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_B ) for some BK(v0)𝐵𝐾subscript𝑣0B\in K(v_{0})italic_B ∈ italic_K ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) and BK(v0)superscript𝐵𝐾superscriptsubscript𝑣0B^{\prime}\in K(v_{0}^{\prime})italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_K ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ). By Lemma 6.4 and by the definition of the W𝑊Witalic_W-action on V𝑉Vitalic_V, wv0=v0𝑤subscript𝑣0superscriptsubscript𝑣0w\!\cdot\!v_{0}=v_{0}^{\prime}italic_w ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT.

Q.E.D.

Let Ksubscript𝐾{\mathcal{B}}_{K}caligraphic_B start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT be the variety of all Borel subgroups of K𝐾Kitalic_K and let K𝐾Kitalic_K act on Ksubscript𝐾{\mathcal{B}}_{K}caligraphic_B start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT by (BK)k:=k-1BKkassignsuperscriptsubscript𝐵𝐾𝑘superscript𝑘1subscript𝐵𝐾𝑘(B_{K})^{k}:=k^{-1}B_{K}k( italic_B start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT := italic_k start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_B start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT italic_k for BKKsubscript𝐵𝐾subscript𝐾B_{K}\in{\mathcal{B}}_{K}italic_B start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT ∈ caligraphic_B start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT and kK𝑘𝐾k\in Kitalic_k ∈ italic_K.

Lemma 6.6.

For any v0V0subscript𝑣0subscript𝑉0v_{0}\in V_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, the map

v0:K(v0)K:BBK,BK(v0):subscriptsubscript𝑣0𝐾subscript𝑣0subscript𝐾:formulae-sequence𝐵𝐵𝐾𝐵𝐾subscript𝑣0{\mathcal{I}}_{v_{0}}:\;\;\;K(v_{0})\longrightarrow{\mathcal{B}}_{K}:\;\;\;B% \longmapsto B\cap K,\hskip 14.454ptB\in K(v_{0})caligraphic_I start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT : italic_K ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) ⟶ caligraphic_B start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT : italic_B ⟼ italic_B ∩ italic_K , italic_B ∈ italic_K ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT )

is a K𝐾Kitalic_K-equivariant isomorphism.

Proof.

By [11, 5.1] (see also [13, Page 113]), BKK𝐵𝐾subscript𝐾B\cap K\in{\mathcal{B}}_{K}italic_B ∩ italic_K ∈ caligraphic_B start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT for every BK(v0)𝐵𝐾subscript𝑣0B\in K(v_{0})italic_B ∈ italic_K ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ). Thus v0subscriptsubscript𝑣0{\mathcal{I}}_{v_{0}}caligraphic_I start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT is well-defined. It is clear that v0subscriptsubscript𝑣0{\mathcal{I}}_{v_{0}}caligraphic_I start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT is K𝐾Kitalic_K-equivariant. To show that v0subscriptsubscript𝑣0{\mathcal{I}}_{v_{0}}caligraphic_I start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT is an isomorphism, we show that v0subscriptsubscript𝑣0{\mathcal{I}}_{v_{0}}caligraphic_I start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT is bijective and that its inverse is an isomorphism from Ksubscript𝐾{\mathcal{B}}_{K}caligraphic_B start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT to K(v0)𝐾subscript𝑣0K(v_{0})italic_K ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ). Fix BK(v0)𝐵𝐾subscript𝑣0B\in K(v_{0})italic_B ∈ italic_K ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) and identify (BK)\KK\𝐵𝐾𝐾subscript𝐾(B\cap K)\backslash K\cong{\mathcal{B}}_{K}( italic_B ∩ italic_K ) \ italic_K ≅ caligraphic_B start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT via

(BK)\KK:(BK)k(BK)k,kK.:\𝐵𝐾𝐾subscript𝐾formulae-sequence𝐵𝐾𝑘superscript𝐵𝐾𝑘𝑘𝐾(B\cap K)\backslash K\longrightarrow{\mathcal{B}}_{K}:\;\;\;(B\cap K)k% \longmapsto(B\cap K)^{k},\hskip 14.454ptk\in K.( italic_B ∩ italic_K ) \ italic_K ⟶ caligraphic_B start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT : ( italic_B ∩ italic_K ) italic_k ⟼ ( italic_B ∩ italic_K ) start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT , italic_k ∈ italic_K .

Consider the action map η:KK(v0):kBk:𝜂𝐾𝐾subscript𝑣0:maps-to𝑘superscript𝐵𝑘\eta:K\to K(v_{0}):k\mapsto B^{k}italic_η : italic_K → italic_K ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) : italic_k ↦ italic_B start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT, kK𝑘𝐾k\in Kitalic_k ∈ italic_K. Then the morphism η~:(BK)\KK(v0):~𝜂\𝐵𝐾𝐾𝐾subscript𝑣0\tilde{\eta}:(B\cap K)\backslash K\to K(v_{0})~ start_ARG italic_η end_ARG : ( italic_B ∩ italic_K ) \ italic_K → italic_K ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) induced by η𝜂\etaitalic_η is the inverse of v0subscriptsubscript𝑣0{\mathcal{I}}_{v_{0}}caligraphic_I start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT. By [5, Proposition 6.7 and Corollary 6.1], η~~𝜂\tilde{\eta}~ start_ARG italic_η end_ARG is an isomorphism if η𝜂\etaitalic_η is separable.

Let 𝔟,𝔨,𝔟𝔨{\mathfrak{b}},{\mathfrak{k}},fraktur_b , fraktur_k , and 𝔤𝔤{\mathfrak{g}}fraktur_g be the Lie algebras of B,K𝐵𝐾B,Kitalic_B , italic_K, and G𝐺Gitalic_G respectively, and let dθ:𝔤𝔤:𝑑𝜃𝔤𝔤d\theta:{\mathfrak{g}}\to{\mathfrak{g}}italic_d italic_θ : fraktur_g → fraktur_g be the differential of θ𝜃\thetaitalic_θ. Since char(𝐤)2char𝐤2{\rm char}({\bf k})\neq 2roman_char ( bold_k ) ≠ 2, θ𝜃\thetaitalic_θ is semisimple on G𝐺Gitalic_G and B𝐵Bitalic_B [15, Section 5.4]. By [15, Theorem 5.4.4(ii)], 𝔨=𝔤dθ={x𝔤:dθ(x)=x}𝔨superscript𝔤𝑑𝜃conditional-set𝑥𝔤𝑑𝜃𝑥𝑥{\mathfrak{k}}={\mathfrak{g}}^{d\theta}=\{x\in{\mathfrak{g}}:d\theta(x)=x\}fraktur_k = fraktur_g start_POSTSUPERSCRIPT italic_d italic_θ end_POSTSUPERSCRIPT = { italic_x ∈ fraktur_g : italic_d italic_θ ( italic_x ) = italic_x }, and the Lie algebra of BK𝐵𝐾B\cap Kitalic_B ∩ italic_K coincides with 𝔟𝔨𝔟𝔨{\mathfrak{b}}\cap{\mathfrak{k}}fraktur_b ∩ fraktur_k. Applying [5, Proposition 6.12] to the quotient morphism GB\G:gBg,gG:𝐺\𝐵𝐺formulae-sequence𝑔superscript𝐵𝑔𝑔𝐺G\to B\backslash G:g\to B^{g},g\in Gitalic_G → italic_B \ italic_G : italic_g → italic_B start_POSTSUPERSCRIPT italic_g end_POSTSUPERSCRIPT , italic_g ∈ italic_G, one sees that η𝜂\etaitalic_η is separable.

Q.E.D.

Identify WK=(K×K)/Ksubscript𝑊𝐾subscript𝐾subscript𝐾𝐾W_{K}=({\mathcal{B}}_{K}\times{\mathcal{B}}_{K})/Kitalic_W start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT = ( caligraphic_B start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT × caligraphic_B start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT ) / italic_K for the diagonal action of K𝐾Kitalic_K on K×Ksubscript𝐾subscript𝐾{\mathcal{B}}_{K}\times{\mathcal{B}}_{K}caligraphic_B start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT × caligraphic_B start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT. For xWK𝑥subscript𝑊𝐾x\in W_{K}italic_x ∈ italic_W start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT, let 𝒪K(x)K×Ksubscript𝒪𝐾𝑥subscript𝐾subscript𝐾{\mathcal{O}}_{K}(x)\subset{\mathcal{B}}_{K}\times{\mathcal{B}}_{K}caligraphic_O start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT ( italic_x ) ⊂ caligraphic_B start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT × caligraphic_B start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT be the corresponding K𝐾Kitalic_K-orbit in K×Ksubscript𝐾subscript𝐾{\mathcal{B}}_{K}\times{\mathcal{B}}_{K}caligraphic_B start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT × caligraphic_B start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT. Let Ksubscript𝐾\leq_{K}≤ start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT denote the Bruhat order on WKsubscript𝑊𝐾W_{K}italic_W start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT.

Let v0,v0V0subscript𝑣0superscriptsubscript𝑣0subscript𝑉0v_{0},v_{0}^{\prime}\in V_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and let

𝒥v0,v0=(v0×v0)-1:K×KK(v0)×K(v0).:subscript𝒥subscript𝑣0superscriptsubscript𝑣0superscriptsubscriptsuperscriptsubscript𝑣0subscriptsubscript𝑣01subscript𝐾subscript𝐾𝐾superscriptsubscript𝑣0𝐾subscript𝑣0\mathcal{J}_{v_{0},v_{0}^{\prime}}=({\mathcal{I}}_{v_{0}^{\prime}}\times{% \mathcal{I}}_{v_{0}})^{-1}:\;\;{\mathcal{B}}_{K}\times{\mathcal{B}}_{K}% \longrightarrow K(v_{0}^{\prime})\times K(v_{0}).caligraphic_J start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT = ( caligraphic_I start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT × caligraphic_I start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT : caligraphic_B start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT × caligraphic_B start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT ⟶ italic_K ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) × italic_K ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) .

For xWK𝑥subscript𝑊𝐾x\in W_{K}italic_x ∈ italic_W start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT, let 𝒪K,v0,v0(x)subscript𝒪𝐾subscript𝑣0superscriptsubscript𝑣0𝑥{\mathcal{O}}_{K,v_{0},v_{0}^{\prime}}(x)caligraphic_O start_POSTSUBSCRIPT italic_K , italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_x ) be the single K𝐾Kitalic_K-orbit in K(v0)×K(v0)𝐾superscriptsubscript𝑣0𝐾subscript𝑣0K(v_{0}^{\prime})\times K(v_{0})italic_K ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) × italic_K ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) given by

(6.6) 𝒪K,v0,v0(x)=𝒥v0,v0(𝒪K(x))K(v0)×K(v0).subscript𝒪𝐾subscript𝑣0superscriptsubscript𝑣0𝑥subscript𝒥subscript𝑣0superscriptsubscript𝑣0subscript𝒪𝐾𝑥𝐾superscriptsubscript𝑣0𝐾subscript𝑣0{\mathcal{O}}_{K,v_{0},v_{0}^{\prime}}(x)=\mathcal{J}_{v_{0},v_{0}^{\prime}}({% \mathcal{O}}_{K}(x))\subset K(v_{0}^{\prime})\times K(v_{0}).caligraphic_O start_POSTSUBSCRIPT italic_K , italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_x ) = caligraphic_J start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( caligraphic_O start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT ( italic_x ) ) ⊂ italic_K ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) × italic_K ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) .

One then has the well-defined map

(6.7) Iv0,v0:WKW:xp(𝒪K,v0,v0(x)),xWK.:subscript𝐼subscript𝑣0superscriptsubscript𝑣0subscript𝑊𝐾𝑊:formulae-sequence𝑥𝑝subscript𝒪𝐾subscript𝑣0superscriptsubscript𝑣0𝑥𝑥subscript𝑊𝐾I_{v_{0},v_{0}^{\prime}}:\;\;\;W_{K}\longrightarrow W:\;\;\;x\longmapsto p% \left({\mathcal{O}}_{K,v_{0},v_{0}^{\prime}}(x)\right),\hskip 14.454ptx\in W_{% K}.italic_I start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT : italic_W start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT ⟶ italic_W : italic_x ⟼ italic_p ( caligraphic_O start_POSTSUBSCRIPT italic_K , italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_x ) ) , italic_x ∈ italic_W start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT .

Let 1111 be the identity element in WKsubscript𝑊𝐾W_{K}italic_W start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT. By definition,

𝒪K,v0,v0(1)={(B,B)K(v0)×K(v0):BK=BK}.subscript𝒪𝐾subscript𝑣0superscriptsubscript𝑣01conditional-setsuperscript𝐵𝐵𝐾superscriptsubscript𝑣0𝐾subscript𝑣0superscript𝐵𝐾𝐵𝐾{\mathcal{O}}_{K,v_{0},v_{0}^{\prime}}(1)=\{(B^{\prime},B)\in K(v_{0}^{\prime}% )\times K(v_{0}):\;B^{\prime}\cap K=B\cap K\}.caligraphic_O start_POSTSUBSCRIPT italic_K , italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( 1 ) = { ( italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_B ) ∈ italic_K ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) × italic_K ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) : italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∩ italic_K = italic_B ∩ italic_K } .
Definition 6.7.

For v0,v0V0subscript𝑣0superscriptsubscript𝑣0subscript𝑉0v_{0},v_{0}^{\prime}\in V_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, let yv0,v0Wθsubscript𝑦subscript𝑣0superscriptsubscript𝑣0superscript𝑊𝜃y_{v_{0},v_{0}^{\prime}}\in W^{\theta}italic_y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ∈ italic_W start_POSTSUPERSCRIPT italic_θ end_POSTSUPERSCRIPT be given by

(6.8) yv0,v0=Iv0,v0(1)=p(B,B),(B,B)𝒪K,v0,v0(1).formulae-sequencesubscript𝑦subscript𝑣0superscriptsubscript𝑣0subscript𝐼subscript𝑣0superscriptsubscript𝑣01𝑝superscript𝐵𝐵for-allsuperscript𝐵𝐵subscript𝒪𝐾subscript𝑣0superscriptsubscript𝑣01y_{v_{0},v_{0}^{\prime}}=I_{v_{0},v_{0}^{\prime}}(1)=p(B^{\prime},B),\hskip 14% .454pt\forall(B^{\prime},B)\in{\mathcal{O}}_{K,v_{0},v_{0}^{\prime}}(1).italic_y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT = italic_I start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( 1 ) = italic_p ( italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_B ) , ∀ ( italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_B ) ∈ caligraphic_O start_POSTSUBSCRIPT italic_K , italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( 1 ) .
Proposition 6.8.

Let v0,v0V0subscript𝑣0superscriptsubscript𝑣0normal-′subscript𝑉0v_{0},v_{0}^{\prime}\in V_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT.

1) The map Iv0,v0subscript𝐼subscript𝑣0superscriptsubscript𝑣0normal-′I_{v_{0},v_{0}^{\prime}}italic_I start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT is a bijection from WKsubscript𝑊𝐾W_{K}italic_W start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT onto {wWθ:wv0=v0}conditional-set𝑤superscript𝑊𝜃normal-⋅𝑤subscript𝑣0superscriptsubscript𝑣0normal-′\{w\in W^{\theta}:\,w\!\cdot\!v_{0}=v_{0}^{\prime}\}{ italic_w ∈ italic_W start_POSTSUPERSCRIPT italic_θ end_POSTSUPERSCRIPT : italic_w ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT };

2) If x1,x2WKsubscript𝑥1subscript𝑥2subscript𝑊𝐾x_{1},x_{2}\in W_{K}italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∈ italic_W start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT are such that x1Kx2subscript𝐾subscript𝑥1subscript𝑥2x_{1}\leq_{K}x_{2}italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≤ start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT, then Iv0,v0(x1)Iv0,v0(x2)subscript𝐼subscript𝑣0superscriptsubscript𝑣0normal-′subscript𝑥1subscript𝐼subscript𝑣0superscriptsubscript𝑣0normal-′subscript𝑥2I_{v_{0},v_{0}^{\prime}}(x_{1})\leq I_{v_{0},v_{0}^{\prime}}(x_{2})italic_I start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ≤ italic_I start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT );

3) yv0,v0Wθsubscript𝑦subscript𝑣0superscriptsubscript𝑣0normal-′superscript𝑊𝜃y_{v_{0},v_{0}^{\prime}}\in W^{\theta}italic_y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ∈ italic_W start_POSTSUPERSCRIPT italic_θ end_POSTSUPERSCRIPT is the unique minimal element in the set {wWθ:wv0=v0}conditional-set𝑤superscript𝑊𝜃normal-⋅𝑤subscript𝑣0superscriptsubscript𝑣0normal-′\{w\in W^{\theta}:\,w\!\cdot\!v_{0}=v_{0}^{\prime}\}{ italic_w ∈ italic_W start_POSTSUPERSCRIPT italic_θ end_POSTSUPERSCRIPT : italic_w ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT } with respect to the Bruhat order on W𝑊Witalic_W.

Proof.

1) By Lemma 6.5 and (6.5), the image of Iv0,v0subscript𝐼subscript𝑣0superscriptsubscript𝑣0I_{v_{0},v_{0}^{\prime}}italic_I start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT is {wWθ:wv0=v0}conditional-set𝑤superscript𝑊𝜃𝑤subscript𝑣0superscriptsubscript𝑣0\{w\in W^{\theta}:\,w\!\cdot\!v_{0}=v_{0}^{\prime}\}{ italic_w ∈ italic_W start_POSTSUPERSCRIPT italic_θ end_POSTSUPERSCRIPT : italic_w ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT }. To show that Iv0,v0subscript𝐼subscript𝑣0superscriptsubscript𝑣0I_{v_{0},v_{0}^{\prime}}italic_I start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT is injective, assume that B,B1K(v0)𝐵subscript𝐵1𝐾subscript𝑣0B,B_{1}\in K(v_{0})italic_B , italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∈ italic_K ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) and B,B1K(v0)superscript𝐵superscriptsubscript𝐵1𝐾superscriptsubscript𝑣0B^{\prime},B_{1}^{\prime}\in K(v_{0}^{\prime})italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_K ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) are such that p(B,B)=p(B1,B1)W𝑝superscript𝐵𝐵𝑝superscriptsubscript𝐵1subscript𝐵1𝑊p(B^{\prime},B)=p(B_{1}^{\prime},B_{1})\in Witalic_p ( italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_B ) = italic_p ( italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ∈ italic_W. We must show that (B,B)superscript𝐵𝐵(B^{\prime},B)( italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_B ) and (B1,B1)superscriptsubscript𝐵1subscript𝐵1(B_{1}^{\prime},B_{1})( italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) are in the same K𝐾Kitalic_K-orbit in K(v0)×K(v0)𝐾superscriptsubscript𝑣0𝐾subscript𝑣0K(v_{0}^{\prime})\times K(v_{0})italic_K ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) × italic_K ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) for the diagonal action of K𝐾Kitalic_K. Without loss of generality, we may assume that B1=Bsubscript𝐵1𝐵B_{1}=Bitalic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_B, and we need to show that B=(B1)ksuperscript𝐵superscriptsuperscriptsubscript𝐵1𝑘B^{\prime}=(B_{1}^{\prime})^{k}italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = ( italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT for some kBK𝑘𝐵𝐾k\in B\cap Kitalic_k ∈ italic_B ∩ italic_K. Let H,H1𝒯θ𝐻subscript𝐻1superscript𝒯𝜃H,H_{1}\in{\mathcal{T}}^{\theta}italic_H , italic_H start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∈ caligraphic_T start_POSTSUPERSCRIPT italic_θ end_POSTSUPERSCRIPT be such that HBB𝐻𝐵superscript𝐵H\subset B\cap B^{\prime}italic_H ⊂ italic_B ∩ italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT and H1BB1subscript𝐻1𝐵superscriptsubscript𝐵1H_{1}\subset B\cap B_{1}^{\prime}italic_H start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⊂ italic_B ∩ italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT. By Lemma 6.1, there exists kBK𝑘𝐵𝐾k\in B\cap Kitalic_k ∈ italic_B ∩ italic_K such that H=(H1)k𝐻superscriptsubscript𝐻1𝑘H=(H_{1})^{k}italic_H = ( italic_H start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT. Thus HBB(B1)k𝐻𝐵superscript𝐵superscriptsuperscriptsubscript𝐵1𝑘H\subset B\cap B^{\prime}\cap(B_{1}^{\prime})^{k}italic_H ⊂ italic_B ∩ italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∩ ( italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT. By the assumption, p(B,B)=p(B1,B)=p((B1)k,Bk)=p((B1)k,B)𝑝superscript𝐵𝐵𝑝superscriptsubscript𝐵1𝐵𝑝superscriptsuperscriptsubscript𝐵1𝑘superscript𝐵𝑘𝑝superscriptsuperscriptsubscript𝐵1𝑘𝐵p(B^{\prime},B)=p(B_{1}^{\prime},B)=p((B_{1}^{\prime})^{k},B^{k})=p((B_{1}^{% \prime})^{k},B)italic_p ( italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_B ) = italic_p ( italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_B ) = italic_p ( ( italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT , italic_B start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ) = italic_p ( ( italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT , italic_B ). Thus B=(B1)ksuperscript𝐵superscriptsuperscriptsubscript𝐵1𝑘B^{\prime}=(B_{1}^{\prime})^{k}italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = ( italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT.

2) Suppose that x1,x2WKsubscript𝑥1subscript𝑥2subscript𝑊𝐾x_{1},x_{2}\in W_{K}italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∈ italic_W start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT are such that x1Kx2subscript𝐾subscript𝑥1subscript𝑥2x_{1}\leq_{K}x_{2}italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≤ start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT. For i=1,2𝑖12i=1,2italic_i = 1 , 2, let wi=Iv0,v0(xi)Wsubscript𝑤𝑖subscript𝐼subscript𝑣0superscriptsubscript𝑣0subscript𝑥𝑖𝑊w_{i}=I_{v_{0},v_{0}^{\prime}}(x_{i})\in Witalic_w start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_I start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) ∈ italic_W, so that 𝒪K,v0,v0(xi)𝒪(wi)subscript𝒪𝐾subscript𝑣0superscriptsubscript𝑣0subscript𝑥𝑖𝒪subscript𝑤𝑖{\mathcal{O}}_{K,v_{0},v_{0}^{\prime}}(x_{i})\subset{\mathcal{O}}(w_{i})caligraphic_O start_POSTSUBSCRIPT italic_K , italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) ⊂ caligraphic_O ( italic_w start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ), where recall from §§\lx@sectionsign§3.4 that 𝒪(wi)𝒪subscript𝑤𝑖{\mathcal{O}}(w_{i})caligraphic_O ( italic_w start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) is the G𝐺Gitalic_G-orbit in ×{\mathcal{B}}\times{\mathcal{B}}caligraphic_B × caligraphic_B corresponding to wiWsubscript𝑤𝑖𝑊w_{i}\in Witalic_w start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ italic_W. For a subset X𝑋Xitalic_X of K×Ksubscript𝐾subscript𝐾{\mathcal{B}}_{K}\times{\mathcal{B}}_{K}caligraphic_B start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT × caligraphic_B start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT (resp. of ×{\mathcal{B}}\times{\mathcal{B}}caligraphic_B × caligraphic_B), let X¯K×Ksuperscript¯𝑋subscript𝐾subscript𝐾\overline{X}^{{\mathcal{B}}_{K}\times{\mathcal{B}}_{K}}¯ start_ARG italic_X end_ARG start_POSTSUPERSCRIPT caligraphic_B start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT × caligraphic_B start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT end_POSTSUPERSCRIPT (resp. X¯×superscript¯𝑋\overline{X}^{{\mathcal{B}}\times{\mathcal{B}}}¯ start_ARG italic_X end_ARG start_POSTSUPERSCRIPT caligraphic_B × caligraphic_B end_POSTSUPERSCRIPT) denote the Zariski closure of X𝑋Xitalic_X in K×Ksubscript𝐾subscript𝐾{\mathcal{B}}_{K}\times{\mathcal{B}}_{K}caligraphic_B start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT × caligraphic_B start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT (resp. in ×{\mathcal{B}}\times{\mathcal{B}}caligraphic_B × caligraphic_B). Since 𝒥v0,v0:K×K×:subscript𝒥subscript𝑣0superscriptsubscript𝑣0subscript𝐾subscript𝐾\mathcal{J}_{v_{0},v_{0}^{\prime}}:{\mathcal{B}}_{K}\times{\mathcal{B}}_{K}\to% {\mathcal{B}}\times{\mathcal{B}}caligraphic_J start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT : caligraphic_B start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT × caligraphic_B start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT → caligraphic_B × caligraphic_B is a morphism, one has

𝒪K,v0,v0(x1)subscript𝒪𝐾subscript𝑣0superscriptsubscript𝑣0subscript𝑥1\displaystyle{\mathcal{O}}_{K,v_{0},v_{0}^{\prime}}(x_{1})caligraphic_O start_POSTSUBSCRIPT italic_K , italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) =𝒥v0,v0(𝒪K(x1))𝒥v0,v0(𝒪K(x1)¯K×K)𝒥v0,v0(𝒪K(x1))¯×absentsubscript𝒥subscript𝑣0superscriptsubscript𝑣0subscript𝒪𝐾subscript𝑥1subscript𝒥subscript𝑣0superscriptsubscript𝑣0superscript¯subscript𝒪𝐾subscript𝑥1subscript𝐾subscript𝐾superscript¯subscript𝒥subscript𝑣0superscriptsubscript𝑣0subscript𝒪𝐾subscript𝑥1\displaystyle=\mathcal{J}_{v_{0},v_{0}^{\prime}}({\mathcal{O}}_{K}(x_{1}))% \subset\mathcal{J}_{v_{0},v_{0}^{\prime}}\left(\overline{{\mathcal{O}}_{K}(x_{% 1})}^{{\mathcal{B}}_{K}\times{\mathcal{B}}_{K}}\right)\subset\overline{% \mathcal{J}_{v_{0},v_{0}^{\prime}}({\mathcal{O}}_{K}(x_{1}))}^{{\mathcal{B}}% \times{\mathcal{B}}}= caligraphic_J start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( caligraphic_O start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ) ⊂ caligraphic_J start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( ¯ start_ARG caligraphic_O start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) end_ARG start_POSTSUPERSCRIPT caligraphic_B start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT × caligraphic_B start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ) ⊂ ¯ start_ARG caligraphic_J start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( caligraphic_O start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ) end_ARG start_POSTSUPERSCRIPT caligraphic_B × caligraphic_B end_POSTSUPERSCRIPT
𝒪(w2)¯×.absentsuperscript¯𝒪subscript𝑤2\displaystyle\subset\overline{{\mathcal{O}}(w_{2})}^{{\mathcal{B}}\times{% \mathcal{B}}}.⊂ ¯ start_ARG caligraphic_O ( italic_w start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) end_ARG start_POSTSUPERSCRIPT caligraphic_B × caligraphic_B end_POSTSUPERSCRIPT .

Thus 𝒪(w1)𝒪(w2)¯×𝒪subscript𝑤1superscript¯𝒪subscript𝑤2{\mathcal{O}}(w_{1})\cap\overline{{\mathcal{O}}(w_{2})}^{{\mathcal{B}}\times{% \mathcal{B}}}\neq\emptysetcaligraphic_O ( italic_w start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ∩ ¯ start_ARG caligraphic_O ( italic_w start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) end_ARG start_POSTSUPERSCRIPT caligraphic_B × caligraphic_B end_POSTSUPERSCRIPT ≠ ∅, and hence w1w2subscript𝑤1subscript𝑤2w_{1}\leq w_{2}italic_w start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≤ italic_w start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT.

3) follows directly from 1) and 2).

Q.E.D.

Remark 6.9.

For v0V0subscript𝑣0subscript𝑉0v_{0}\in V_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, let WK(v0)={wW:wv0=v0}subscript𝑊𝐾subscript𝑣0conditional-set𝑤𝑊𝑤subscript𝑣0subscript𝑣0W_{K}(v_{0})=\{w\in W:w\!\cdot\!v_{0}=v_{0}\}italic_W start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) = { italic_w ∈ italic_W : italic_w ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT }. Then for v0,v0V0subscript𝑣0superscriptsubscript𝑣0subscript𝑉0v_{0},v_{0}^{\prime}\in V_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, the set {wWθ:wv0=v0}conditional-set𝑤superscript𝑊𝜃𝑤subscript𝑣0superscriptsubscript𝑣0\{w\in W^{\theta}:\,w\!\cdot\!v_{0}=v_{0}^{\prime}\}{ italic_w ∈ italic_W start_POSTSUPERSCRIPT italic_θ end_POSTSUPERSCRIPT : italic_w ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT } coincides with the coset yv0,v0WK(v0)subscript𝑦subscript𝑣0superscriptsubscript𝑣0subscript𝑊𝐾subscript𝑣0y_{v_{0},v_{0}^{\prime}}W_{K}(v_{0})italic_y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_W start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) in Wθsuperscript𝑊𝜃W^{\theta}italic_W start_POSTSUPERSCRIPT italic_θ end_POSTSUPERSCRIPT. It is easy to see that Iv0,v0:WKWK(v0):subscript𝐼subscript𝑣0subscript𝑣0subscript𝑊𝐾subscript𝑊𝐾subscript𝑣0I_{v_{0},v_{0}}:W_{K}\to W_{K}(v_{0})italic_I start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT : italic_W start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT → italic_W start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) is a group isomorphism (see, for example, [12, Proposition 2.8]). Hence, by 3) of Proposition 6.8, every coset in Wθ/WK(v0)superscript𝑊𝜃subscript𝑊𝐾subscript𝑣0W^{\theta}/W_{K}(v_{0})italic_W start_POSTSUPERSCRIPT italic_θ end_POSTSUPERSCRIPT / italic_W start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) has a unique minimal length representative. In case K𝐾Kitalic_K is disconnected, this last assertion is no longer true in the case when G=PGL(4)𝐺𝑃𝐺𝐿4G=PGL(4)italic_G = italic_P italic_G italic_L ( 4 ) and K𝐾Kitalic_K has connected component of the identity equal to the image of GL(2)×GL(2)𝐺𝐿2𝐺𝐿2GL(2)\times GL(2)italic_G italic_L ( 2 ) × italic_G italic_L ( 2 ) in G𝐺Gitalic_G.

6.9. The involution θv:HcanHcan:subscript𝜃𝑣subscript𝐻cansubscript𝐻can\theta_{v}:H_{\rm can}\to H_{\rm can}italic_θ start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT : italic_H start_POSTSUBSCRIPT roman_can end_POSTSUBSCRIPT → italic_H start_POSTSUBSCRIPT roman_can end_POSTSUBSCRIPT

Let vV𝑣𝑉v\in Vitalic_v ∈ italic_V. Choose any (B,H)𝒞θ𝐵𝐻subscript𝒞𝜃(B,H)\in{\mathcal{C}}_{\theta}( italic_B , italic_H ) ∈ caligraphic_C start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT such that BK(v)𝐵𝐾𝑣B\in K(v)italic_B ∈ italic_K ( italic_v ), and let θ|H:HH:evaluated-at𝜃𝐻𝐻𝐻\theta|_{H}:H\to Hitalic_θ | start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT : italic_H → italic_H be the restriction of θ𝜃\thetaitalic_θ to H𝐻Hitalic_H. Define

θv:=TB,Hθ|HTB,H-1:HcanHcan:[B,H,h][B,H,θ(h)],hH.:assignsubscript𝜃𝑣evaluated-atsubscript𝑇𝐵𝐻𝜃𝐻superscriptsubscript𝑇𝐵𝐻1subscript𝐻cansubscript𝐻can:formulae-sequence𝐵𝐻𝐵𝐻𝜃𝐻\theta_{v}:=T_{B,H}\circ\theta|_{H}\circ T_{B,H}^{-1}:\;H_{\rm can}% \longrightarrow H_{\rm can}:\;\;[B,H,h]\longmapsto[B,H,\theta(h)],\hskip 14.45% 4pth\in H.italic_θ start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT := italic_T start_POSTSUBSCRIPT italic_B , italic_H end_POSTSUBSCRIPT ∘ italic_θ | start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT ∘ italic_T start_POSTSUBSCRIPT italic_B , italic_H end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT : italic_H start_POSTSUBSCRIPT roman_can end_POSTSUBSCRIPT ⟶ italic_H start_POSTSUBSCRIPT roman_can end_POSTSUBSCRIPT : [ italic_B , italic_H , italic_h ] ⟼ [ italic_B , italic_H , italic_θ ( italic_h ) ] , italic_h ∈ italic_H .

For another (B,H)𝒞θsuperscript𝐵superscript𝐻subscript𝒞𝜃(B^{\prime},H^{\prime})\in{\mathcal{C}}_{\theta}( italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_H start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ∈ caligraphic_C start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT such that BK(v)superscript𝐵𝐾𝑣B^{\prime}\in K(v)italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_K ( italic_v ), there exists kK𝑘𝐾k\in Kitalic_k ∈ italic_K such that B=Bksuperscript𝐵superscript𝐵𝑘B^{\prime}=B^{k}italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_B start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT and H=Hksuperscript𝐻superscript𝐻𝑘H^{\prime}=H^{k}italic_H start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_H start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT, and for any hH𝐻h\in Hitalic_h ∈ italic_H, [B,H,h]=[Bk,Hk,k-1hk]Hcan𝐵𝐻superscript𝐵𝑘superscript𝐻𝑘superscript𝑘1𝑘subscript𝐻can[B,H,h]=[B^{k},H^{k},k^{-1}hk]\in H_{\rm can}[ italic_B , italic_H , italic_h ] = [ italic_B start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT , italic_H start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT , italic_k start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_h italic_k ] ∈ italic_H start_POSTSUBSCRIPT roman_can end_POSTSUBSCRIPT, and

[Bk,Hk,θ(k-1hk)]=[Bk,Hk,k-1θ(h)k]=[B,H,θ(h)].superscript𝐵𝑘superscript𝐻𝑘𝜃superscript𝑘1𝑘superscript𝐵𝑘superscript𝐻𝑘superscript𝑘1𝜃𝑘𝐵𝐻𝜃[B^{k},H^{k},\theta(k^{-1}hk)]=[B^{k},H^{k},k^{-1}\theta(h)k]=[B,H,\theta(h)].[ italic_B start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT , italic_H start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT , italic_θ ( italic_k start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_h italic_k ) ] = [ italic_B start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT , italic_H start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT , italic_k start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_θ ( italic_h ) italic_k ] = [ italic_B , italic_H , italic_θ ( italic_h ) ] .

Thus θv:HcanHcan:subscript𝜃𝑣subscript𝐻cansubscript𝐻can\theta_{v}:H_{\rm can}\to H_{\rm can}italic_θ start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT : italic_H start_POSTSUBSCRIPT roman_can end_POSTSUBSCRIPT → italic_H start_POSTSUBSCRIPT roman_can end_POSTSUBSCRIPT is independent of the choice of (B,H)𝒞θ𝐵𝐻subscript𝒞𝜃(B,H)\in{\mathcal{C}}_{\theta}( italic_B , italic_H ) ∈ caligraphic_C start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT. By (6.1),

(6.9) θv=ϕ(v)θ:HcanHcan.:subscript𝜃𝑣italic-ϕ𝑣𝜃subscript𝐻cansubscript𝐻can\theta_{v}=\phi(v)\theta:\;\;\;H_{\rm can}\longrightarrow H_{\rm can}.italic_θ start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT = italic_ϕ ( italic_v ) italic_θ : italic_H start_POSTSUBSCRIPT roman_can end_POSTSUBSCRIPT ⟶ italic_H start_POSTSUBSCRIPT roman_can end_POSTSUBSCRIPT .

The induced involution on the set ΔΔ\Deltaroman_Δ of canonical roots of G𝐺Gitalic_G (see §§\lx@sectionsign§6.2) will also be denoted by θvsubscript𝜃𝑣\theta_{v}italic_θ start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT.

6.10. The subsets p(sα,v)𝑝subscript𝑠𝛼𝑣p(s_{\alpha},v)italic_p ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT , italic_v ) and root types

Let αΓ𝛼Γ\alpha\in\Gammaitalic_α ∈ roman_Γ. A parabolic subgroup of G𝐺Gitalic_G is said to be of type α𝛼\alphaitalic_α if it is of the form BBsαB𝐵𝐵subscript𝑠𝛼𝐵B\cup Bs_{\alpha}Bitalic_B ∪ italic_B italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT italic_B for some B𝐵B\in{\mathcal{B}}italic_B ∈ caligraphic_B. Let 𝒫αsubscript𝒫𝛼{\mathcal{P}}_{\alpha}caligraphic_P start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT be the variety of all parabolic subgroups of G𝐺Gitalic_G of type α𝛼\alphaitalic_α. The action of G𝐺Gitalic_G on 𝒫αsubscript𝒫𝛼{\mathcal{P}}_{\alpha}caligraphic_P start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT by conjugation is transitive, and one has the G𝐺Gitalic_G-equivariant surjective morphism

πα:𝒫α:πα(B)=B(BsαB),B.:subscript𝜋𝛼subscript𝒫𝛼:formulae-sequencesubscript𝜋𝛼𝐵𝐵𝐵subscript𝑠𝛼𝐵𝐵\pi_{\alpha}:\;\;{\mathcal{B}}\longrightarrow{\mathcal{P}}_{\alpha}:\;\;\pi_{% \alpha}(B)=B\cup(Bs_{\alpha}B),\hskip 14.454ptB\in{\mathcal{B}}.italic_π start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT : caligraphic_B ⟶ caligraphic_P start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT : italic_π start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ( italic_B ) = italic_B ∪ ( italic_B italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT italic_B ) , italic_B ∈ caligraphic_B .

For vV𝑣𝑉v\in Vitalic_v ∈ italic_V, let

p(sα,v)={vV:K(v)πα-1(πα(K(v))}.fragmentspfragments(subscript𝑠𝛼,v)fragments{superscript𝑣V:Kfragments(superscript𝑣)superscriptsubscript𝜋𝛼1fragments(subscript𝜋𝛼fragments(Kfragments(v))}.p(s_{\alpha},v)=\{v^{\prime}\in V:\;K(v^{\prime})\subset\pi_{\alpha}^{-1}(\pi_% {\alpha}(K(v))\}.italic_p ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT , italic_v ) = { italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_V : italic_K ( italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ⊂ italic_π start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_π start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ( italic_K ( italic_v ) ) } .

It is well-known (see, for example, [13, §§\lx@sectionsign§2.4] and [14]) that for each αΓ𝛼Γ\alpha\in\Gammaitalic_α ∈ roman_Γ and vV𝑣𝑉v\in Vitalic_v ∈ italic_V, the subset p(sα,v)𝑝subscript𝑠𝛼𝑣p(s_{\alpha},v)italic_p ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT , italic_v ) of V𝑉Vitalic_V has either one or two or three elements, depending on the type of α𝛼\alphaitalic_α relative to v𝑣vitalic_v, and that m(sα)v𝑚subscript𝑠𝛼𝑣m(s_{\alpha})\!\cdot\!vitalic_m ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ) ⋅ italic_v is the unique maximal element in p(sα,v)𝑝subscript𝑠𝛼𝑣p(s_{\alpha},v)italic_p ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT , italic_v ) with respect to the Bruhat order on V𝑉Vitalic_V.

The case analysis of p(sα,v)𝑝subscript𝑠𝛼𝑣p(s_{\alpha},v)italic_p ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT , italic_v ) and the definition of the type of α𝛼\alphaitalic_α relative to v𝑣vitalic_v given in [13, §§\lx@sectionsign§2.4] and [14] make use of a standard pair (B0,H0)𝒞subscript𝐵0subscript𝐻0𝒞(B_{0},H_{0})\in{\mathcal{C}}( italic_B start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) ∈ caligraphic_C, but the results are independent of the choice of (B0,H0)subscript𝐵0subscript𝐻0(B_{0},H_{0})( italic_B start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ). Based on the results from [13, §§\lx@sectionsign§2.4] and [14], we give the equivalent definitions of root types in Definition 6.10 and summarize the results on p(sα,v)𝑝subscript𝑠𝛼𝑣p(s_{\alpha},v)italic_p ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT , italic_v ) from [13, §§\lx@sectionsign§2.4] and [14] in the following Proposition 6.11.

Definition 6.10.

Let vV𝑣𝑉v\in Vitalic_v ∈ italic_V. An αΓ𝛼Γ\alpha\in\Gammaitalic_α ∈ roman_Γ is said to be imaginary (resp. real, complex) for v𝑣vitalic_v if θv(α)=αsubscript𝜃𝑣𝛼𝛼\theta_{v}(\alpha)=\alphaitalic_θ start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT ( italic_α ) = italic_α (resp. θv(α)=-α,θv(α)±αformulae-sequencesubscript𝜃𝑣𝛼𝛼subscript𝜃𝑣𝛼plus-or-minus𝛼\theta_{v}(\alpha)=-\alpha,\;\theta_{v}(\alpha)\neq\pm\alphaitalic_θ start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT ( italic_α ) = - italic_α , italic_θ start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT ( italic_α ) ≠ ± italic_α). A simple imaginary root α𝛼\alphaitalic_α for vV𝑣𝑉v\in Vitalic_v ∈ italic_V is said to be compact if m(sα)v=v𝑚subscript𝑠𝛼𝑣𝑣m(s_{\alpha})\!\cdot\!v=vitalic_m ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ) ⋅ italic_v = italic_v and non-compact if m(sα)vv𝑚subscript𝑠𝛼𝑣𝑣m(s_{\alpha})\!\cdot\!v\neq vitalic_m ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ) ⋅ italic_v ≠ italic_v. A simple non-compact imaginary root α𝛼\alphaitalic_α is said to be cancellative if sαv=vsubscript𝑠𝛼𝑣𝑣s_{\alpha}\!\cdot\!v=vitalic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ⋅ italic_v = italic_v and non-cancellative if sαvvsubscript𝑠𝛼𝑣𝑣s_{\alpha}\!\cdot\!v\neq vitalic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ⋅ italic_v ≠ italic_v (see [17, §§\lx@sectionsign§2.4]). A simple real root α𝛼\alphaitalic_α for vV𝑣𝑉v\in Vitalic_v ∈ italic_V is said to be cancellative if p(sα,v)𝑝subscript𝑠𝛼𝑣p(s_{\alpha},v)italic_p ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT , italic_v ) has two elements and non-cancellative if p(sα,v)𝑝subscript𝑠𝛼𝑣p(s_{\alpha},v)italic_p ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT , italic_v ) has three elements. We will use the following notation.

Ivcsubscriptsuperscript𝐼𝑐𝑣\displaystyle I^{c}_{v}italic_I start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT ={αΓ:αis compact imaginary forv},absentconditional-set𝛼Γ𝛼is compact imaginary for𝑣\displaystyle=\{\alpha\in\Gamma:\;\alpha\;\mbox{is compact imaginary for}\,v\},= { italic_α ∈ roman_Γ : italic_α is compact imaginary for italic_v } ,
Ivn,=superscriptsubscript𝐼𝑣𝑛\displaystyle I_{v}^{n,=}italic_I start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n , = end_POSTSUPERSCRIPT ={αΓ:αis non-compact imaginary and cancellative forv},absentconditional-set𝛼Γ𝛼is non-compact imaginary and cancellative for𝑣\displaystyle=\{\alpha\in\Gamma:\;\alpha\;\mbox{is non-compact imaginary and % cancellative for}\,v\},= { italic_α ∈ roman_Γ : italic_α is non-compact imaginary and cancellative for italic_v } ,
Ivn,superscriptsubscript𝐼𝑣𝑛\displaystyle I_{v}^{n,\neq}italic_I start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n , ≠ end_POSTSUPERSCRIPT ={αΓ:αis non-compact imaginary and non-cancellative forv},absentconditional-set𝛼Γ𝛼is non-compact imaginary and non-cancellative for𝑣\displaystyle=\{\alpha\in\Gamma:\;\alpha\;\mbox{is non-compact imaginary and % non-cancellative for}\,v\},= { italic_α ∈ roman_Γ : italic_α is non-compact imaginary and non-cancellative for italic_v } ,
Rv=superscriptsubscript𝑅𝑣\displaystyle R_{v}^{=}italic_R start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT = end_POSTSUPERSCRIPT ={αΓ:αis real and cancellative forv},absentconditional-set𝛼Γ𝛼is real and cancellative for𝑣\displaystyle=\{\alpha\in\Gamma:\;\alpha\;\mbox{is real and cancellative for}% \,v\},= { italic_α ∈ roman_Γ : italic_α is real and cancellative for italic_v } ,
Rvsuperscriptsubscript𝑅𝑣\displaystyle R_{v}^{\neq}italic_R start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ≠ end_POSTSUPERSCRIPT ={αΓ:αis real and non-cancellative forv},absentconditional-set𝛼Γ𝛼is real and non-cancellative for𝑣\displaystyle=\{\alpha\in\Gamma:\;\alpha\;\mbox{is real and non-cancellative % for}\,v\},= { italic_α ∈ roman_Γ : italic_α is real and non-cancellative for italic_v } ,
Cv+superscriptsubscript𝐶𝑣\displaystyle C_{v}^{+}italic_C start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ={αΓ:αis complex forvandθv(α)Δ+},absentconditional-set𝛼Γ𝛼is complex for𝑣andsubscript𝜃𝑣𝛼superscriptΔ\displaystyle=\{\alpha\in\Gamma:\;\alpha\;\mbox{is complex for}\,v\;\mbox{and}% \;\theta_{v}(\alpha)\in\Delta^{+}\},= { italic_α ∈ roman_Γ : italic_α is complex for italic_v and italic_θ start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT ( italic_α ) ∈ roman_Δ start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT } ,
Cv-superscriptsubscript𝐶𝑣\displaystyle C_{v}^{-}italic_C start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT ={αΓ:αis complex forvandθv(α)-Δ+}.absentconditional-set𝛼Γ𝛼is complex for𝑣andsubscript𝜃𝑣𝛼superscriptΔ\displaystyle=\{\alpha\in\Gamma:\;\alpha\;\mbox{is complex for}\,v\;\mbox{and}% \;\theta_{v}(\alpha)\in-\Delta^{+}\}.= { italic_α ∈ roman_Γ : italic_α is complex for italic_v and italic_θ start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT ( italic_α ) ∈ - roman_Δ start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT } .

We also set Ivn=Ivn,=Ivn,superscriptsubscript𝐼𝑣𝑛superscriptsubscript𝐼𝑣𝑛superscriptsubscript𝐼𝑣𝑛I_{v}^{n}=I_{v}^{n,=}\cup I_{v}^{n,\neq}italic_I start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT = italic_I start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n , = end_POSTSUPERSCRIPT ∪ italic_I start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n , ≠ end_POSTSUPERSCRIPT, Iv=IvcIvnsubscript𝐼𝑣superscriptsubscript𝐼𝑣𝑐superscriptsubscript𝐼𝑣𝑛I_{v}=I_{v}^{c}\cup I_{v}^{n}italic_I start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT = italic_I start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT ∪ italic_I start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT, and Rv=Rv=Rvsubscript𝑅𝑣superscriptsubscript𝑅𝑣superscriptsubscript𝑅𝑣R_{v}=R_{v}^{=}\cup R_{v}^{\neq}italic_R start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT = italic_R start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT = end_POSTSUPERSCRIPT ∪ italic_R start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ≠ end_POSTSUPERSCRIPT.

Proposition 6.11.

Let vV𝑣𝑉v\in Vitalic_v ∈ italic_V and αΓ𝛼normal-Γ\alpha\in\Gammaitalic_α ∈ roman_Γ.

Case 1), αIvc𝛼superscriptsubscript𝐼𝑣𝑐\alpha\in I_{v}^{c}italic_α ∈ italic_I start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT. Then p(sα,v)={v}𝑝subscript𝑠𝛼𝑣𝑣p(s_{\alpha},v)=\{v\}italic_p ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT , italic_v ) = { italic_v }.

Case 2), αIvn,=𝛼superscriptsubscript𝐼𝑣𝑛\alpha\in I_{v}^{n,=}italic_α ∈ italic_I start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n , = end_POSTSUPERSCRIPT. Then sαv=vm(sα)vnormal-⋅subscript𝑠𝛼𝑣𝑣normal-⋅𝑚subscript𝑠𝛼𝑣s_{\alpha}\!\cdot\!v=v\neq m(s_{\alpha})\!\cdot\!vitalic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ⋅ italic_v = italic_v ≠ italic_m ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ) ⋅ italic_v, and p(sα,v)={v,m(sα)v}𝑝subscript𝑠𝛼𝑣𝑣normal-⋅𝑚subscript𝑠𝛼𝑣p(s_{\alpha},v)=\{v,\;m(s_{\alpha})\!\cdot\!v\}italic_p ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT , italic_v ) = { italic_v , italic_m ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ) ⋅ italic_v }. Moreover, αRm(sα)v=𝛼superscriptsubscript𝑅normal-⋅𝑚subscript𝑠𝛼𝑣\alpha\in R_{m(s_{\alpha})\cdot v}^{=}italic_α ∈ italic_R start_POSTSUBSCRIPT italic_m ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ) ⋅ italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT = end_POSTSUPERSCRIPT.

Case 3), αIvn,𝛼superscriptsubscript𝐼𝑣𝑛\alpha\in I_{v}^{n,\neq}italic_α ∈ italic_I start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n , ≠ end_POSTSUPERSCRIPT. Then v𝑣vitalic_v, sαvnormal-⋅subscript𝑠𝛼𝑣s_{\alpha}\!\cdot\!vitalic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ⋅ italic_v, and m(sα)vnormal-⋅𝑚subscript𝑠𝛼𝑣m(s_{\alpha})\!\cdot\!vitalic_m ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ) ⋅ italic_v are pair-wise distinct, and p(sα,v)={v,sαv,m(sα)v}𝑝subscript𝑠𝛼𝑣𝑣normal-⋅subscript𝑠𝛼𝑣normal-⋅𝑚subscript𝑠𝛼𝑣p(s_{\alpha},v)=\{v,\;s_{\alpha}\!\cdot\!v,\;m(s_{\alpha})\!\cdot\!v\}italic_p ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT , italic_v ) = { italic_v , italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ⋅ italic_v , italic_m ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ) ⋅ italic_v }. Moreover, αIsαvn,𝛼superscriptsubscript𝐼normal-⋅subscript𝑠𝛼𝑣𝑛\alpha\in I_{s_{\alpha}\cdot v}^{n,\neq}italic_α ∈ italic_I start_POSTSUBSCRIPT italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ⋅ italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n , ≠ end_POSTSUPERSCRIPT and αRm(sα)v𝛼superscriptsubscript𝑅normal-⋅𝑚subscript𝑠𝛼𝑣\alpha\in R_{m(s_{\alpha})\cdot v}^{\neq}italic_α ∈ italic_R start_POSTSUBSCRIPT italic_m ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ) ⋅ italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ≠ end_POSTSUPERSCRIPT.

Case 4), αRv=.𝛼superscriptsubscript𝑅𝑣\alpha\in R_{v}^{=}.italic_α ∈ italic_R start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT = end_POSTSUPERSCRIPT . Then there exists vp(sα,v)superscript𝑣normal-′𝑝subscript𝑠𝛼𝑣v^{\prime}\in p(s_{\alpha},v)italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_p ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT , italic_v ), vvsuperscript𝑣normal-′𝑣v^{\prime}\neq vitalic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≠ italic_v, such that sαv=v,v=m(sα)v=sαvformulae-sequencenormal-⋅subscript𝑠𝛼superscript𝑣normal-′superscript𝑣normal-′𝑣normal-⋅𝑚subscript𝑠𝛼superscript𝑣normal-′normal-⋅subscript𝑠𝛼𝑣s_{\alpha}\!\cdot\!v^{\prime}=v^{\prime},\,v=m(s_{\alpha})\!\cdot\!v^{\prime}=% s_{\alpha}\!\cdot\!vitalic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ⋅ italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_v = italic_m ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ) ⋅ italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ⋅ italic_v, and p(sα,v)={v,v}𝑝subscript𝑠𝛼𝑣superscript𝑣normal-′𝑣p(s_{\alpha},v)=\{v^{\prime},\;v\}italic_p ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT , italic_v ) = { italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_v }. Moreover, αIvn,=𝛼superscriptsubscript𝐼superscript𝑣normal-′𝑛\alpha\in I_{v^{\prime}}^{n,=}italic_α ∈ italic_I start_POSTSUBSCRIPT italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n , = end_POSTSUPERSCRIPT.

Case 5), αRv𝛼superscriptsubscript𝑅𝑣\alpha\in R_{v}^{\neq}italic_α ∈ italic_R start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ≠ end_POSTSUPERSCRIPT. Then there exists vp(sα,v)superscript𝑣normal-′𝑝subscript𝑠𝛼𝑣v^{\prime}\in p(s_{\alpha},v)italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_p ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT , italic_v ) such that v,sαvsuperscript𝑣normal-′normal-⋅subscript𝑠𝛼superscript𝑣normal-′v^{\prime},\,s_{\alpha}\!\cdot\!v^{\prime}italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ⋅ italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT and v=m(sα)v=sαv𝑣normal-⋅𝑚subscript𝑠𝛼superscript𝑣normal-′normal-⋅subscript𝑠𝛼𝑣v=m(s_{\alpha})\!\cdot\!v^{\prime}=s_{\alpha}\!\cdot\!vitalic_v = italic_m ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ) ⋅ italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ⋅ italic_v are pair-wise distinct, and p(sα,v)={v,sαv,v}𝑝subscript𝑠𝛼𝑣superscript𝑣normal-′normal-⋅subscript𝑠𝛼superscript𝑣normal-′𝑣p(s_{\alpha},v)=\{v^{\prime},\,s_{\alpha}\!\cdot\!v^{\prime},\,v\}italic_p ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT , italic_v ) = { italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ⋅ italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_v }. Moreover, αIvn,Isαvn,𝛼superscriptsubscript𝐼superscript𝑣normal-′𝑛superscriptsubscript𝐼normal-⋅subscript𝑠𝛼superscript𝑣normal-′𝑛\alpha\in I_{v^{\prime}}^{n,\neq}\cap I_{s_{\alpha}\cdot v^{\prime}}^{n,\neq}italic_α ∈ italic_I start_POSTSUBSCRIPT italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n , ≠ end_POSTSUPERSCRIPT ∩ italic_I start_POSTSUBSCRIPT italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ⋅ italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n , ≠ end_POSTSUPERSCRIPT.

Case 6), αCv+.𝛼superscriptsubscript𝐶𝑣\alpha\in C_{v}^{+}.italic_α ∈ italic_C start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT . Then m(sα)v=sαvvnormal-⋅𝑚subscript𝑠𝛼𝑣normal-⋅subscript𝑠𝛼𝑣𝑣m(s_{\alpha})\!\cdot\!v=s_{\alpha}\!\cdot\!v\neq vitalic_m ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ) ⋅ italic_v = italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ⋅ italic_v ≠ italic_v, and p(sα,v)={v,m(sα)v}𝑝subscript𝑠𝛼𝑣𝑣normal-⋅𝑚subscript𝑠𝛼𝑣p(s_{\alpha},v)=\{v,\,m(s_{\alpha})\!\cdot\!v\}italic_p ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT , italic_v ) = { italic_v , italic_m ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ) ⋅ italic_v }. Moreover, αCm(sα)v-𝛼subscriptsuperscript𝐶normal-⋅𝑚subscript𝑠𝛼𝑣\alpha\in C^{-}_{m(s_{\alpha})\cdot v}italic_α ∈ italic_C start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ) ⋅ italic_v end_POSTSUBSCRIPT.

Case 7), αCv-.𝛼superscriptsubscript𝐶𝑣\alpha\in C_{v}^{-}.italic_α ∈ italic_C start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT . Then m(sα)v=vsαvnormal-⋅𝑚subscript𝑠𝛼𝑣𝑣normal-⋅subscript𝑠𝛼𝑣m(s_{\alpha})\!\cdot\!v=v\neq s_{\alpha}\!\cdot\!vitalic_m ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ) ⋅ italic_v = italic_v ≠ italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ⋅ italic_v, and p(sα,v)={v,sαv}𝑝subscript𝑠𝛼𝑣𝑣normal-⋅subscript𝑠𝛼𝑣p(s_{\alpha},v)=\{v,\,s_{\alpha}\!\cdot\!v\}italic_p ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT , italic_v ) = { italic_v , italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ⋅ italic_v }. Moreover, αCsαv+𝛼subscriptsuperscript𝐶normal-⋅subscript𝑠𝛼𝑣\alpha\in C^{+}_{s_{\alpha}\cdot v}italic_α ∈ italic_C start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ⋅ italic_v end_POSTSUBSCRIPT.

Lemma 6.12.

[13, Page 122] Let v,vV𝑣superscript𝑣normal-′𝑉v,v^{\prime}\in Vitalic_v , italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_V and αΓ𝛼normal-Γ\alpha\in\Gammaitalic_α ∈ roman_Γ. Then vp(sα,v)superscript𝑣normal-′𝑝subscript𝑠𝛼𝑣v^{\prime}\in p(s_{\alpha},v)italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_p ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT , italic_v ) if and only if m(sα)v=m(sα)vnormal-⋅𝑚subscript𝑠𝛼superscript𝑣normal-′normal-⋅𝑚subscript𝑠𝛼𝑣m(s_{\alpha})\!\cdot\!v^{\prime}=m(s_{\alpha})\!\cdot\!vitalic_m ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ) ⋅ italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_m ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ) ⋅ italic_v. Moreover, p(sα,v)=p(sα,v)𝑝subscript𝑠𝛼superscript𝑣normal-′𝑝subscript𝑠𝛼𝑣p(s_{\alpha},v^{\prime})=p(s_{\alpha},v)italic_p ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT , italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) = italic_p ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT , italic_v ) for all vp(sα,v).superscript𝑣normal-′𝑝subscript𝑠𝛼𝑣v^{\prime}\in p(s_{\alpha},v).italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_p ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT , italic_v ) .

Lemma 6.13.

[13, §§\lx@sectionsign§3.2] For αΓ𝛼normal-Γ\alpha\in\Gammaitalic_α ∈ roman_Γ and vV𝑣𝑉v\in Vitalic_v ∈ italic_V, m(sα)vvnormal-⋅𝑚subscript𝑠𝛼𝑣𝑣m(s_{\alpha})\!\cdot\!v\neq vitalic_m ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ) ⋅ italic_v ≠ italic_v if and only if αCv+Ivn𝛼superscriptsubscript𝐶𝑣superscriptsubscript𝐼𝑣𝑛\alpha\in C_{v}^{+}\cup I_{v}^{n}italic_α ∈ italic_C start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ∪ italic_I start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT. If αCv+𝛼superscriptsubscript𝐶𝑣\alpha\in C_{v}^{+}italic_α ∈ italic_C start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT, then ϕ(m(sα)v)=sαϕ(v)sθ(α)italic-ϕnormal-⋅𝑚subscript𝑠𝛼𝑣subscript𝑠𝛼italic-ϕ𝑣subscript𝑠𝜃𝛼\phi(m(s_{\alpha})\!\cdot\!v)=s_{\alpha}\phi(v)s_{\theta(\alpha)}italic_ϕ ( italic_m ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ) ⋅ italic_v ) = italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT italic_ϕ ( italic_v ) italic_s start_POSTSUBSCRIPT italic_θ ( italic_α ) end_POSTSUBSCRIPT, and l(ϕ(m(sα)v))=l(ϕ(v))+2𝑙italic-ϕnormal-⋅𝑚subscript𝑠𝛼𝑣𝑙italic-ϕ𝑣2l(\phi(m(s_{\alpha})\!\cdot\!v))=l(\phi(v))+2italic_l ( italic_ϕ ( italic_m ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ) ⋅ italic_v ) ) = italic_l ( italic_ϕ ( italic_v ) ) + 2. If αIvn𝛼superscriptsubscript𝐼𝑣𝑛\alpha\in I_{v}^{n}italic_α ∈ italic_I start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT, then ϕ(m(sα)v)=sαϕ(v)=ϕ(v)sθ(α)>ϕ(v)italic-ϕnormal-⋅𝑚subscript𝑠𝛼𝑣subscript𝑠𝛼italic-ϕ𝑣italic-ϕ𝑣subscript𝑠𝜃𝛼italic-ϕ𝑣\phi(m(s_{\alpha})\!\cdot\!v)=s_{\alpha}\phi(v)=\phi(v)s_{\theta(\alpha)}>\phi% (v)italic_ϕ ( italic_m ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ) ⋅ italic_v ) = italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT italic_ϕ ( italic_v ) = italic_ϕ ( italic_v ) italic_s start_POSTSUBSCRIPT italic_θ ( italic_α ) end_POSTSUBSCRIPT > italic_ϕ ( italic_v ) and ϕ(sαv)=ϕ(v)italic-ϕnormal-⋅subscript𝑠𝛼𝑣italic-ϕ𝑣\phi(s_{\alpha}\!\cdot\!v)=\phi(v)italic_ϕ ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ⋅ italic_v ) = italic_ϕ ( italic_v ).

6.11. Reduced decompositions and subexpressions

We refer to [12, §§\lx@sectionsign§5-7] and [13, §§\lx@sectionsign§4] for more detail on this subsection.

Definition 6.14.

[13, Definition 3.2.3] Let vV𝑣𝑉v\in Vitalic_v ∈ italic_V. A reduced decomposition of v𝑣vitalic_v is a pair (𝐯,𝐬)𝐯𝐬({\bf v},{\bf s})( bold_v , bold_s ), where 𝐯=(v0,v1,,vk)𝐯subscript𝑣0subscript𝑣1subscript𝑣𝑘{\bf v}=(v_{0},v_{1},\ldots,v_{k})bold_v = ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_v start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) is a sequence in V𝑉Vitalic_V and 𝐬=(sα1,,sαk)𝐬subscript𝑠subscript𝛼1subscript𝑠subscript𝛼𝑘{\bf s}=(s_{\alpha_{1}},\ldots,s_{\alpha_{k}})bold_s = ( italic_s start_POSTSUBSCRIPT italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT , … , italic_s start_POSTSUBSCRIPT italic_α start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) is a sequence of simple reflections in W𝑊Witalic_W, such that v0V0subscript𝑣0subscript𝑉0v_{0}\in V_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, vk=vsubscript𝑣𝑘𝑣v_{k}=vitalic_v start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = italic_v, and for each j[1,k]𝑗1𝑘j\in[1,k]italic_j ∈ [ 1 , italic_k ],

αjCvj-1+Ivj-1nandvj=m(sαj)vj-1.subscript𝛼𝑗superscriptsubscript𝐶subscript𝑣𝑗1superscriptsubscript𝐼subscript𝑣𝑗1𝑛andsubscript𝑣𝑗𝑚subscript𝑠subscript𝛼𝑗subscript𝑣𝑗1\alpha_{j}\in C_{v_{j-1}}^{+}\cup I_{v_{j-1}}^{n}\;\;\;\mbox{and}\;\;\;v_{j}=m% (s_{\alpha_{j}})\!\cdot\!v_{j-1}.italic_α start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∈ italic_C start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT italic_j - 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ∪ italic_I start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT italic_j - 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT and italic_v start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = italic_m ( italic_s start_POSTSUBSCRIPT italic_α start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) ⋅ italic_v start_POSTSUBSCRIPT italic_j - 1 end_POSTSUBSCRIPT .

The integer k𝑘kitalic_k is called the length of the reduced decomposition (𝐯,𝐬)𝐯𝐬({\bf v},{\bf s})( bold_v , bold_s ).

Every vV𝑣𝑉v\in Vitalic_v ∈ italic_V has a reduced decomposition and all reduced decompositions of v𝑣vitalic_v have the same length, which will be denoted by l(v)𝑙𝑣l(v)italic_l ( italic_v ) and called the length of v𝑣vitalic_v (see [13, 3.2 and Page 113]).

Definition 6.15.

[13, Definition 4.3] Let vV𝑣𝑉v\in Vitalic_v ∈ italic_V and let (𝐯=(v0,v1,,vk),𝐬=(sα1,,sαk))formulae-sequence𝐯subscript𝑣0subscript𝑣1subscript𝑣𝑘𝐬subscript𝑠subscript𝛼1subscript𝑠subscript𝛼𝑘({\bf v}=(v_{0},v_{1},\ldots,v_{k}),\,{\bf s}=(s_{\alpha_{1}},\ldots,s_{\alpha% _{k}}))( bold_v = ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_v start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) , bold_s = ( italic_s start_POSTSUBSCRIPT italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT , … , italic_s start_POSTSUBSCRIPT italic_α start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) ) be a reduced decomposition for v𝑣vitalic_v. A subexpression of (𝐯,𝐬)𝐯𝐬({\bf v},{\bf s})( bold_v , bold_s ) is a sequence 𝐮=(u0,u1,,uk)𝐮subscript𝑢0subscript𝑢1subscript𝑢𝑘{\bf u}=(u_{0},u_{1},\ldots,u_{k})bold_u = ( italic_u start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_u start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_u start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) in V𝑉Vitalic_V such that u0=v0subscript𝑢0subscript𝑣0u_{0}=v_{0}italic_u start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and one of the following holds for each j[1,k]𝑗1𝑘j\in[1,k]italic_j ∈ [ 1 , italic_k ]:

Case 1), uj=uj-1subscript𝑢𝑗subscript𝑢𝑗1u_{j}=u_{j-1}italic_u start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = italic_u start_POSTSUBSCRIPT italic_j - 1 end_POSTSUBSCRIPT;

Case 2), αjCuj-1+Iuj-1nsubscript𝛼𝑗superscriptsubscript𝐶subscript𝑢𝑗1superscriptsubscript𝐼subscript𝑢𝑗1𝑛\alpha_{j}\in C_{u_{j-1}}^{+}\cup I_{u_{j-1}}^{n}italic_α start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∈ italic_C start_POSTSUBSCRIPT italic_u start_POSTSUBSCRIPT italic_j - 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ∪ italic_I start_POSTSUBSCRIPT italic_u start_POSTSUBSCRIPT italic_j - 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT and uj=m(sαj)uj-1subscript𝑢𝑗𝑚subscript𝑠subscript𝛼𝑗subscript𝑢𝑗1u_{j}=m(s_{\alpha_{j}})\!\cdot\!u_{j-1}italic_u start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = italic_m ( italic_s start_POSTSUBSCRIPT italic_α start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) ⋅ italic_u start_POSTSUBSCRIPT italic_j - 1 end_POSTSUBSCRIPT;

Case 3), αjIuj-1n,subscript𝛼𝑗superscriptsubscript𝐼subscript𝑢𝑗1𝑛\alpha_{j}\in I_{u_{j-1}}^{n,\neq}italic_α start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∈ italic_I start_POSTSUBSCRIPT italic_u start_POSTSUBSCRIPT italic_j - 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n , ≠ end_POSTSUPERSCRIPT and uj=sαjuj-1subscript𝑢𝑗subscript𝑠subscript𝛼𝑗subscript𝑢𝑗1u_{j}=s_{\alpha_{j}}\!\cdot\!u_{j-1}italic_u start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = italic_s start_POSTSUBSCRIPT italic_α start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⋅ italic_u start_POSTSUBSCRIPT italic_j - 1 end_POSTSUBSCRIPT.

In this case, uksubscript𝑢𝑘u_{k}italic_u start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT is called the final term of the subexpression 𝐮=(u0,u1,,uk)𝐮subscript𝑢0subscript𝑢1subscript𝑢𝑘{\bf u}=(u_{0},u_{1},\ldots,u_{k})bold_u = ( italic_u start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_u start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_u start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ).

Proposition 6.16.

[13, Proposition 4.4] Let v,vV𝑣superscript𝑣normal-′𝑉v,v^{\prime}\in Vitalic_v , italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_V and let (𝐯,𝐬)𝐯𝐬({\bf v},{\bf s})( bold_v , bold_s ) be a reduced decomposition for v𝑣vitalic_v. Then vvsuperscript𝑣normal-′𝑣v^{\prime}\leq vitalic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≤ italic_v if and only if there exists a subexpression of (𝐯,𝐬)𝐯𝐬({\bf v},{\bf s})( bold_v , bold_s ) with final term vsuperscript𝑣normal-′v^{\prime}italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT.

7. Analysis on Yv0(v)subscript𝑌subscript𝑣0𝑣Y_{v_{0}}(v)italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) and proof of Theorem 2.2

7.1. The set Yv0subscript𝑌subscript𝑣0Y_{v_{0}}italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT

For v0V0subscript𝑣0subscript𝑉0v_{0}\in V_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, let Yv0=vVYv0(v)subscript𝑌subscript𝑣0subscript𝑣𝑉subscript𝑌subscript𝑣0𝑣Y_{v_{0}}=\bigcup_{v\in V}Y_{v_{0}}(v)italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT = ⋃ start_POSTSUBSCRIPT italic_v ∈ italic_V end_POSTSUBSCRIPT italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ). Recall that l𝑙litalic_l denotes both the length function on W𝑊Witalic_W and the length function on V𝑉Vitalic_V.

Lemma 7.1.

Let v0V0subscript𝑣0subscript𝑉0v_{0}\in V_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and yW𝑦𝑊y\in Witalic_y ∈ italic_W. Then yYv0𝑦subscript𝑌subscript𝑣0y\in Y_{v_{0}}italic_y ∈ italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT if and only if l(m(y)v0)=l(y)𝑙normal-⋅𝑚𝑦subscript𝑣0𝑙𝑦l(m(y)\!\cdot\!v_{0})=l(y)italic_l ( italic_m ( italic_y ) ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) = italic_l ( italic_y ).

Proof.

It is clear that l(m(y)v0)l(y)𝑙𝑚𝑦subscript𝑣0𝑙𝑦l(m(y)\!\cdot\!v_{0})\leq l(y)italic_l ( italic_m ( italic_y ) ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) ≤ italic_l ( italic_y ) for any yW𝑦𝑊y\in Witalic_y ∈ italic_W. Assume first that yYv0(v)𝑦subscript𝑌subscript𝑣0𝑣y\in Y_{v_{0}}(v)italic_y ∈ italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) for some vV𝑣𝑉v\in Vitalic_v ∈ italic_V. If l(m(y)v0)<l(y)𝑙𝑚𝑦subscript𝑣0𝑙𝑦l(m(y)\!\cdot\!v_{0})<l(y)italic_l ( italic_m ( italic_y ) ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) < italic_l ( italic_y ), then there exists y1<ysubscript𝑦1𝑦y_{1}<yitalic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT < italic_y such that m(y1)v0=m(y)v0𝑚subscript𝑦1subscript𝑣0𝑚𝑦subscript𝑣0m(y_{1})\!\cdot\!v_{0}=m(y)\!\cdot\!v_{0}italic_m ( italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_m ( italic_y ) ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, so y1Wv0(v)subscript𝑦1subscript𝑊subscript𝑣0𝑣y_{1}\in W_{v_{0}}(v)italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∈ italic_W start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ), which is a contradiction. Thus l(m(y)v0)=l(y)𝑙𝑚𝑦subscript𝑣0𝑙𝑦l(m(y)\!\cdot\!v_{0})=l(y)italic_l ( italic_m ( italic_y ) ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) = italic_l ( italic_y ). Conversely, assume that l(m(y)v0)=l(y)𝑙𝑚𝑦subscript𝑣0𝑙𝑦l(m(y)\!\cdot\!v_{0})=l(y)italic_l ( italic_m ( italic_y ) ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) = italic_l ( italic_y ). Let v=m(y)v0𝑣𝑚𝑦subscript𝑣0v=m(y)\!\cdot\!v_{0}italic_v = italic_m ( italic_y ) ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT. Then yYv0(v)𝑦subscript𝑌subscript𝑣0𝑣y\in Y_{v_{0}}(v)italic_y ∈ italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ).

Q.E.D.

Lemma-Notation 7.2.

Let yW𝑦𝑊y\in Witalic_y ∈ italic_W. If y=sαksα1𝑦subscript𝑠subscript𝛼𝑘normal-⋯subscript𝑠subscript𝛼1y=s_{\alpha_{k}}\cdots s_{\alpha_{1}}italic_y = italic_s start_POSTSUBSCRIPT italic_α start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⋯ italic_s start_POSTSUBSCRIPT italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT is a reduced word of y𝑦yitalic_y, the sequence 𝐬=(sα1,,sαk)𝐬subscript𝑠subscript𝛼1normal-…subscript𝑠subscript𝛼𝑘{\bf s}=(s_{\alpha_{1}},\ldots,s_{\alpha_{k}})bold_s = ( italic_s start_POSTSUBSCRIPT italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT , … , italic_s start_POSTSUBSCRIPT italic_α start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) is also called a reduced word of y-1superscript𝑦1y^{-1}italic_y start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT. Let

R(y-1)={𝐬:𝐬is a reduced word ofy-1}.𝑅superscript𝑦1conditional-set𝐬𝐬is a reduced word ofsuperscript𝑦1R(y^{-1})=\{{\bf s}:\;{\bf s}\;\mbox{is a reduced word of}\;y^{-1}\}.italic_R ( italic_y start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ) = { bold_s : bold_s is a reduced word of italic_y start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT } .

Let v0V0subscript𝑣0subscript𝑉0v_{0}\in V_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and yYv0𝑦subscript𝑌subscript𝑣0y\in Y_{v_{0}}italic_y ∈ italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT. For 𝐬=(sα1,,sαk)R(y-1)𝐬subscript𝑠subscript𝛼1normal-…subscript𝑠subscript𝛼𝑘𝑅superscript𝑦1{\bf s}=(s_{\alpha_{1}},\ldots,s_{\alpha_{k}})\in R(y^{-1})bold_s = ( italic_s start_POSTSUBSCRIPT italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT , … , italic_s start_POSTSUBSCRIPT italic_α start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) ∈ italic_R ( italic_y start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ), let

(7.1) 𝐯v0(𝐬)=(v0,m(y1)v0,,m(yk)v0),subscript𝐯subscript𝑣0𝐬subscript𝑣0𝑚subscript𝑦1subscript𝑣0𝑚subscript𝑦𝑘subscript𝑣0{\bf v}_{v_{0}}({\bf s})=(v_{0},\;m(y_{1})\!\cdot\!v_{0},\;\cdots,\;m(y_{k})\!% \cdot\!v_{0}),bold_v start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( bold_s ) = ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_m ( italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , ⋯ , italic_m ( italic_y start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) ,

where yj=sαjsα1subscript𝑦𝑗subscript𝑠subscript𝛼𝑗normal-⋯subscript𝑠subscript𝛼1y_{j}=s_{\alpha_{j}}\cdots s_{\alpha_{1}}italic_y start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = italic_s start_POSTSUBSCRIPT italic_α start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⋯ italic_s start_POSTSUBSCRIPT italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT for j[1,k]𝑗1𝑘j\in[1,k]italic_j ∈ [ 1 , italic_k ]. Then (𝐯v0(𝐬),𝐬)subscript𝐯subscript𝑣0𝐬𝐬({\bf v}_{v_{0}}({\bf s}),{\bf s})( bold_v start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( bold_s ) , bold_s ) is a reduced decomposition for m(y)v0normal-⋅𝑚𝑦subscript𝑣0m(y)\!\cdot\!v_{0}italic_m ( italic_y ) ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, which will be called the reduced decomposition of m(y)v0normal-⋅𝑚𝑦subscript𝑣0m(y)\!\cdot\!v_{0}italic_m ( italic_y ) ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT associated to 𝐬𝐬{\bf s}bold_s.

Proof.

If (𝐯v0(𝐬),𝐬)subscript𝐯subscript𝑣0𝐬𝐬({\bf v}_{v_{0}}({\bf s}),{\bf s})( bold_v start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( bold_s ) , bold_s ) is not a reduced decomposition for m(y)v0𝑚𝑦subscript𝑣0m(y)\!\cdot\!v_{0}italic_m ( italic_y ) ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, then l(m(y)v0)<l(y)𝑙𝑚𝑦subscript𝑣0𝑙𝑦l(m(y)\!\cdot\!v_{0})<l(y)italic_l ( italic_m ( italic_y ) ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) < italic_l ( italic_y ), and so yYv0𝑦subscript𝑌subscript𝑣0y\not\in Y_{v_{0}}italic_y ∉ italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT by Lemma 7.1.

Q.E.D.

7.2. Local analysis of Yv0(v)subscript𝑌subscript𝑣0𝑣Y_{v_{0}}(v)italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ), Part I

Recall the monoidal operation \ast on W𝑊Witalic_W in §§\lx@sectionsign§3.6.

Lemma 7.3.

Let v0V0subscript𝑣0subscript𝑉0v_{0}\in V_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, vV𝑣𝑉v\in Vitalic_v ∈ italic_V, and αΓ𝛼normal-Γ\alpha\in\Gammaitalic_α ∈ roman_Γ.

1) If wWv0(m(sα)v)𝑤subscript𝑊subscript𝑣0normal-⋅𝑚subscript𝑠𝛼𝑣w\in W_{v_{0}}(m(s_{\alpha})\!\cdot\!v)italic_w ∈ italic_W start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_m ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ) ⋅ italic_v ), then wWv0(v)𝑤subscript𝑊subscript𝑣0superscript𝑣normal-′w\in W_{v_{0}}(v^{\prime})italic_w ∈ italic_W start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) for all vp(sα,v)superscript𝑣normal-′𝑝subscript𝑠𝛼𝑣v^{\prime}\in p(s_{\alpha},v)italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_p ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT , italic_v );

2) If wWv0(v)𝑤subscript𝑊subscript𝑣0𝑣w\in W_{v_{0}}(v)italic_w ∈ italic_W start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ), then sαwWv0(v)normal-∗subscript𝑠𝛼𝑤subscript𝑊subscript𝑣0superscript𝑣normal-′s_{\alpha}\ast w\in W_{v_{0}}(v^{\prime})italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ∗ italic_w ∈ italic_W start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) for all vp(sα,v)superscript𝑣normal-′𝑝subscript𝑠𝛼𝑣v^{\prime}\in p(s_{\alpha},v)italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_p ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT , italic_v ).

Proof.

1) follows from the fact that vm(sα)vsuperscript𝑣𝑚subscript𝑠𝛼𝑣v^{\prime}\leq m(s_{\alpha})\!\cdot\!vitalic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≤ italic_m ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ) ⋅ italic_v for all vp(sα,v)superscript𝑣𝑝subscript𝑠𝛼𝑣v^{\prime}\in p(s_{\alpha},v)italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_p ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT , italic_v ).

2). Assume that wWv0(v)𝑤subscript𝑊subscript𝑣0𝑣w\in W_{v_{0}}(v)italic_w ∈ italic_W start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ). Then by Lemma 3.5,

m(sα)vm(sα)m(w)v0=m(sαw)v0.𝑚subscript𝑠𝛼𝑣𝑚subscript𝑠𝛼𝑚𝑤subscript𝑣0𝑚subscript𝑠𝛼𝑤subscript𝑣0m(s_{\alpha})\!\cdot\!v\leq m(s_{\alpha})\!\cdot\!m(w)\!\cdot\!v_{0}=m(s_{% \alpha}\ast w)\!\cdot\!v_{0}.italic_m ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ) ⋅ italic_v ≤ italic_m ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ) ⋅ italic_m ( italic_w ) ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_m ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ∗ italic_w ) ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT .

Thus sαwWv0(m(sα)v)subscript𝑠𝛼𝑤subscript𝑊subscript𝑣0𝑚subscript𝑠𝛼𝑣s_{\alpha}\ast w\in W_{v_{0}}(m(s_{\alpha})\!\cdot\!v)italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ∗ italic_w ∈ italic_W start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_m ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ) ⋅ italic_v ). 2) now follows from 1).

Q.E.D.

The following Lemma 7.4 is the key in proving Theorem 2.2.

Lemma 7.4.

Let v0V0subscript𝑣0subscript𝑉0v_{0}\in V_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, vV𝑣𝑉v\in Vitalic_v ∈ italic_V, and yYv0(v)𝑦subscript𝑌subscript𝑣0𝑣y\in Y_{v_{0}}(v)italic_y ∈ italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ). Let αΓ𝛼normal-Γ\alpha\in\Gammaitalic_α ∈ roman_Γ and yWsuperscript𝑦normal-′𝑊y^{\prime}\in Witalic_y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_W be such that y=sαy>y𝑦subscript𝑠𝛼superscript𝑦normal-′superscript𝑦normal-′y=s_{\alpha}y^{\prime}>y^{\prime}italic_y = italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT italic_y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT > italic_y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT. Then there exists up(sα,v)\{v,m(sα)v}𝑢normal-\𝑝subscript𝑠𝛼𝑣𝑣normal-⋅𝑚subscript𝑠𝛼𝑣u\in p(s_{\alpha},\,v)\backslash\{v,\,m(s_{\alpha})\!\cdot\!v\}italic_u ∈ italic_p ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT , italic_v ) \ { italic_v , italic_m ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ) ⋅ italic_v } such that yYv0(u)superscript𝑦normal-′subscript𝑌subscript𝑣0𝑢y^{\prime}\in Y_{v_{0}}(u)italic_y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_u ). Moreover, there are three possibilities:

1) αCu+𝛼superscriptsubscript𝐶𝑢\alpha\in C_{u}^{+}italic_α ∈ italic_C start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT and v=m(sα)u𝑣normal-⋅𝑚subscript𝑠𝛼𝑢v=m(s_{\alpha})\!\cdot\!uitalic_v = italic_m ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ) ⋅ italic_u. In this case, αCv-𝛼superscriptsubscript𝐶𝑣\alpha\in C_{v}^{-}italic_α ∈ italic_C start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT;

2) αIvn𝛼superscriptsubscript𝐼𝑣𝑛\alpha\in I_{v}^{n}italic_α ∈ italic_I start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT and v=m(sα)u𝑣normal-⋅𝑚subscript𝑠𝛼𝑢v=m(s_{\alpha})\!\cdot\!uitalic_v = italic_m ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ) ⋅ italic_u. In this case, αRv𝛼subscript𝑅𝑣\alpha\in R_{v}italic_α ∈ italic_R start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT;

3) αIun,𝛼superscriptsubscript𝐼𝑢𝑛\alpha\in I_{u}^{n,\neq}italic_α ∈ italic_I start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n , ≠ end_POSTSUPERSCRIPT and v=sαu𝑣normal-⋅subscript𝑠𝛼𝑢v=s_{\alpha}\!\cdot\!uitalic_v = italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ⋅ italic_u. In this case, αIvn,𝛼superscriptsubscript𝐼𝑣𝑛\alpha\in I_{v}^{n,\neq}italic_α ∈ italic_I start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n , ≠ end_POSTSUPERSCRIPT.

Proof.

Let y=sαk-1sα1superscript𝑦subscript𝑠subscript𝛼𝑘1subscript𝑠subscript𝛼1y^{\prime}=s_{\alpha_{k-1}}\cdots s_{\alpha_{1}}italic_y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_s start_POSTSUBSCRIPT italic_α start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⋯ italic_s start_POSTSUBSCRIPT italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT be a reduced word of ysuperscript𝑦y^{\prime}italic_y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT. Then

𝐬=(sα1,,sαk-1,sαk)R(y-1),𝐬subscript𝑠subscript𝛼1subscript𝑠subscript𝛼𝑘1subscript𝑠subscript𝛼𝑘𝑅superscript𝑦1{\bf s}=(s_{\alpha_{1}},\ldots,s_{\alpha_{k-1}},s_{\alpha_{k}})\in R(y^{-1}),bold_s = ( italic_s start_POSTSUBSCRIPT italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT , … , italic_s start_POSTSUBSCRIPT italic_α start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT , italic_s start_POSTSUBSCRIPT italic_α start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) ∈ italic_R ( italic_y start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ) ,

where αk=αsubscript𝛼𝑘𝛼\alpha_{k}=\alphaitalic_α start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = italic_α. Consider the reduced decomposition (𝐯v0(𝐬),𝐬)subscript𝐯subscript𝑣0𝐬𝐬({\bf v}_{v_{0}}({\bf s}),{\bf s})( bold_v start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( bold_s ) , bold_s ) of m(y)v0𝑚𝑦subscript𝑣0m(y)\!\cdot\!v_{0}italic_m ( italic_y ) ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT. By Proposition 6.16, there is a subexpression 𝐮=(u0=v0,u1,,,,uk-1,uk)fragmentsufragments(subscript𝑢0subscript𝑣0,subscript𝑢1,,,,subscript𝑢𝑘1,subscript𝑢𝑘){\bf u}=(u_{0}=v_{0},\,u_{1},,\ldots,,u_{k-1},\,u_{k})bold_u = ( italic_u start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_u start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , , … , , italic_u start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT , italic_u start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) of (𝐯v0(𝐬),𝐬)subscript𝐯subscript𝑣0𝐬𝐬({\bf v}_{v_{0}}({\bf s}),{\bf s})( bold_v start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( bold_s ) , bold_s ) with final term uk=vsubscript𝑢𝑘𝑣u_{k}=vitalic_u start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = italic_v, and we let u=uk-1𝑢subscript𝑢𝑘1u=u_{k-1}italic_u = italic_u start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT. Since y<ysuperscript𝑦𝑦y^{\prime}<yitalic_y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT < italic_y and um(y)v0𝑢𝑚superscript𝑦subscript𝑣0u\leq m(y^{\prime})\!\cdot\!v_{0}italic_u ≤ italic_m ( italic_y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT by Proposition 6.16, uv𝑢𝑣u\neq vitalic_u ≠ italic_v and um(sα)u=m(sα)v𝑢𝑚subscript𝑠𝛼𝑢𝑚subscript𝑠𝛼𝑣u\neq m(s_{\alpha})\!\cdot\!u=m(s_{\alpha})\!\cdot\!vitalic_u ≠ italic_m ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ) ⋅ italic_u = italic_m ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ) ⋅ italic_v. By the definition of a subexpression, one has either 1), 2), or 3). By Proposition 6.11, αCv-𝛼superscriptsubscript𝐶𝑣\alpha\in C_{v}^{-}italic_α ∈ italic_C start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT in 1), αRv𝛼subscript𝑅𝑣\alpha\in R_{v}italic_α ∈ italic_R start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT in 2), and αIvn,𝛼superscriptsubscript𝐼𝑣𝑛\alpha\in I_{v}^{n,\neq}italic_α ∈ italic_I start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n , ≠ end_POSTSUPERSCRIPT in 3).

It remains to show that yYv0(u)superscript𝑦subscript𝑌subscript𝑣0𝑢y^{\prime}\in Y_{v_{0}}(u)italic_y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_u ). Since um(y)v0𝑢𝑚superscript𝑦subscript𝑣0u\leq m(y^{\prime})\!\cdot\!v_{0}italic_u ≤ italic_m ( italic_y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, yWv0(u)superscript𝑦subscript𝑊subscript𝑣0𝑢y^{\prime}\in W_{v_{0}}(u)italic_y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_W start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_u ). Suppose that yYv0(u)superscript𝑦subscript𝑌subscript𝑣0𝑢y^{\prime}\notin Y_{v_{0}}(u)italic_y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∉ italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_u ). Let y′′Wv0(u)superscript𝑦′′subscript𝑊subscript𝑣0𝑢y^{\prime\prime}\in W_{v_{0}}(u)italic_y start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ∈ italic_W start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_u ) be such that y′′<ysuperscript𝑦′′superscript𝑦y^{\prime\prime}<y^{\prime}italic_y start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT < italic_y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT. By Lemma 7.3, sαy′′Wv0(v)subscript𝑠𝛼superscript𝑦′′subscript𝑊subscript𝑣0𝑣s_{\alpha}\ast y^{\prime\prime}\in W_{v_{0}}(v)italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ∗ italic_y start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ∈ italic_W start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ). Since sαy′′sαy=ysubscript𝑠𝛼superscript𝑦′′subscript𝑠𝛼superscript𝑦𝑦s_{\alpha}\ast y^{\prime\prime}\leq s_{\alpha}\ast y^{\prime}=yitalic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ∗ italic_y start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ≤ italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ∗ italic_y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_y and l(sαy′′)1+l(y′′)<1+l(y)=l(y)𝑙subscript𝑠𝛼superscript𝑦′′1𝑙superscript𝑦′′1𝑙superscript𝑦𝑙𝑦l(s_{\alpha}\ast y^{\prime\prime})\leq 1+l(y^{\prime\prime})<1+l(y^{\prime})=l% (y)italic_l ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ∗ italic_y start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) ≤ 1 + italic_l ( italic_y start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) < 1 + italic_l ( italic_y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) = italic_l ( italic_y ), we have sαy′′<ysubscript𝑠𝛼superscript𝑦′′𝑦s_{\alpha}\ast y^{\prime\prime}<yitalic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ∗ italic_y start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT < italic_y, which is a contradiction. Thus yYv0(u)superscript𝑦subscript𝑌subscript𝑣0𝑢y^{\prime}\in Y_{v_{0}}(u)italic_y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_u ).

Q.E.D.

7.3. Proof of Theorem 2.2

Assume that v,vV𝑣superscript𝑣𝑉v,v^{\prime}\in Vitalic_v , italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_V are such that ϕ(v)=ϕ(v)italic-ϕ𝑣italic-ϕsuperscript𝑣\phi(v)=\phi(v^{\prime})italic_ϕ ( italic_v ) = italic_ϕ ( italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) and that there exists yYv0(v)Yv0(v)𝑦subscript𝑌subscript𝑣0𝑣subscript𝑌subscript𝑣0superscript𝑣y\in Y_{v_{0}}(v)\cap Y_{v_{0}}(v^{\prime})italic_y ∈ italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) ∩ italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ). We use induction on l(y)𝑙𝑦l(y)italic_l ( italic_y ) to show that v=v𝑣superscript𝑣v=v^{\prime}italic_v = italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT.

If l(y)=0𝑙𝑦0l(y)=0italic_l ( italic_y ) = 0, then y=1𝑦1y=1italic_y = 1. By Lemma 4.4, v=v=v0𝑣superscript𝑣subscript𝑣0v=v^{\prime}=v_{0}italic_v = italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT. Assume now that l(y)1𝑙𝑦1l(y)\geq 1italic_l ( italic_y ) ≥ 1, and choose αΓ𝛼Γ\alpha\in\Gammaitalic_α ∈ roman_Γ such that y:=sαy<yassignsuperscript𝑦subscript𝑠𝛼𝑦𝑦y^{\prime}:=s_{\alpha}y<yitalic_y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT := italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT italic_y < italic_y. By Lemma 7.4, there exist up(sα,v)𝑢𝑝subscript𝑠𝛼𝑣u\in p(s_{\alpha},v)italic_u ∈ italic_p ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT , italic_v ) and up(sα,v)superscript𝑢𝑝subscript𝑠𝛼superscript𝑣u^{\prime}\in p(s_{\alpha},v^{\prime})italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_p ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT , italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) such that yYv0(u)Yv0(u)superscript𝑦subscript𝑌subscript𝑣0𝑢subscript𝑌subscript𝑣0superscript𝑢y^{\prime}\in Y_{v_{0}}(u)\cap Y_{v_{0}}(u^{\prime})italic_y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_u ) ∩ italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ), and

α(Cv-RvIvn,)(Cv-RvIvn,).𝛼superscriptsubscript𝐶𝑣subscript𝑅𝑣superscriptsubscript𝐼𝑣𝑛superscriptsubscript𝐶superscript𝑣subscript𝑅superscript𝑣superscriptsubscript𝐼superscript𝑣𝑛\alpha\in\left(C_{v}^{-}\cup R_{v}\cup I_{v}^{n,\neq}\right)\cap\left(C_{v^{% \prime}}^{-}\cup R_{v^{\prime}}\cup I_{v^{\prime}}^{n,\neq}\right).italic_α ∈ ( italic_C start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT ∪ italic_R start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT ∪ italic_I start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n , ≠ end_POSTSUPERSCRIPT ) ∩ ( italic_C start_POSTSUBSCRIPT italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT ∪ italic_R start_POSTSUBSCRIPT italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ∪ italic_I start_POSTSUBSCRIPT italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n , ≠ end_POSTSUPERSCRIPT ) .

Since ϕ(v)=ϕ(v)italic-ϕ𝑣italic-ϕsuperscript𝑣\phi(v)=\phi(v^{\prime})italic_ϕ ( italic_v ) = italic_ϕ ( italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ), one has by (6.9) that θv(α)=θv(α)subscript𝜃𝑣𝛼subscript𝜃superscript𝑣𝛼\theta_{v}(\alpha)=\theta_{v^{\prime}}(\alpha)italic_θ start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT ( italic_α ) = italic_θ start_POSTSUBSCRIPT italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_α ). Thus αCv-𝛼superscriptsubscript𝐶𝑣\alpha\in C_{v}^{-}italic_α ∈ italic_C start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT (resp. Rv,Ivn,subscript𝑅𝑣superscriptsubscript𝐼𝑣𝑛R_{v},\,I_{v}^{n,\neq}italic_R start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT , italic_I start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n , ≠ end_POSTSUPERSCRIPT) if and only if αCv-𝛼superscriptsubscript𝐶superscript𝑣\alpha\in C_{v^{\prime}}^{-}italic_α ∈ italic_C start_POSTSUBSCRIPT italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT (resp. Rv,Ivn,subscript𝑅superscript𝑣superscriptsubscript𝐼superscript𝑣𝑛R_{v^{\prime}},\;I_{v^{\prime}}^{n,\neq}italic_R start_POSTSUBSCRIPT italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT , italic_I start_POSTSUBSCRIPT italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n , ≠ end_POSTSUPERSCRIPT). We now look at the cases separately.

Case 1): αCv-𝛼superscriptsubscript𝐶𝑣\alpha\in C_{v}^{-}italic_α ∈ italic_C start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT. In this case, αCv-𝛼superscriptsubscript𝐶superscript𝑣\alpha\in C_{v^{\prime}}^{-}italic_α ∈ italic_C start_POSTSUBSCRIPT italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT, v=m(sα)u=sαu𝑣𝑚subscript𝑠𝛼𝑢subscript𝑠𝛼𝑢v=m(s_{\alpha})\!\cdot\!u=s_{\alpha}\!\cdot\!uitalic_v = italic_m ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ) ⋅ italic_u = italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ⋅ italic_u, and v=m(sα)u=sαusuperscript𝑣𝑚subscript𝑠𝛼superscript𝑢subscript𝑠𝛼superscript𝑢v^{\prime}=m(s_{\alpha})\!\cdot\!u^{\prime}=s_{\alpha}\!\cdot\!u^{\prime}italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_m ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ) ⋅ italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ⋅ italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT. By Lemma 6.13, ϕ(u)=sαϕ(v)θ(sα)=sαϕ(v)θ(sα)=ϕ(u).italic-ϕ𝑢subscript𝑠𝛼italic-ϕ𝑣𝜃subscript𝑠𝛼subscript𝑠𝛼italic-ϕsuperscript𝑣𝜃subscript𝑠𝛼italic-ϕsuperscript𝑢\phi(u)=s_{\alpha}\phi(v)\theta(s_{\alpha})=s_{\alpha}\phi(v^{\prime})\theta(s% _{\alpha})=\phi(u^{\prime}).italic_ϕ ( italic_u ) = italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT italic_ϕ ( italic_v ) italic_θ ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ) = italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT italic_ϕ ( italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) italic_θ ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ) = italic_ϕ ( italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) . By the induction assumption, u=u𝑢superscript𝑢u=u^{\prime}italic_u = italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT. Thus v=sαu=sαu=v𝑣subscript𝑠𝛼𝑢subscript𝑠𝛼superscript𝑢superscript𝑣v=s_{\alpha}\!\cdot\!u=s_{\alpha}\!\cdot\!u^{\prime}=v^{\prime}italic_v = italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ⋅ italic_u = italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ⋅ italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT.

Case 2): αRv𝛼subscript𝑅𝑣\alpha\in R_{v}italic_α ∈ italic_R start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT. In this case, αRv𝛼subscript𝑅superscript𝑣\alpha\in R_{v^{\prime}}italic_α ∈ italic_R start_POSTSUBSCRIPT italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT, v=m(sα)u𝑣𝑚subscript𝑠𝛼𝑢v=m(s_{\alpha})\!\cdot\!uitalic_v = italic_m ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ) ⋅ italic_u, and v=m(sα)usuperscript𝑣𝑚subscript𝑠𝛼superscript𝑢v^{\prime}=m(s_{\alpha})\!\cdot\!u^{\prime}italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_m ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ) ⋅ italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT. By Lemma 6.13, ϕ(u)=sαϕ(v)=sαϕ(v)=ϕ(u)italic-ϕ𝑢subscript𝑠𝛼italic-ϕ𝑣subscript𝑠𝛼italic-ϕsuperscript𝑣italic-ϕsuperscript𝑢\phi(u)=s_{\alpha}\phi(v)=s_{\alpha}\phi(v^{\prime})=\phi(u^{\prime})italic_ϕ ( italic_u ) = italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT italic_ϕ ( italic_v ) = italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT italic_ϕ ( italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) = italic_ϕ ( italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ). By the induction assumption, u=u𝑢superscript𝑢u=u^{\prime}italic_u = italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT. Thus v=m(sα)u=m(sα)u=v𝑣𝑚subscript𝑠𝛼𝑢𝑚subscript𝑠𝛼superscript𝑢superscript𝑣v=m(s_{\alpha})\!\cdot\!u=m(s_{\alpha})\!\cdot\!u^{\prime}=v^{\prime}italic_v = italic_m ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ) ⋅ italic_u = italic_m ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ) ⋅ italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT.

Case 3): αIvn,𝛼superscriptsubscript𝐼𝑣𝑛\alpha\in I_{v}^{n,\neq}italic_α ∈ italic_I start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n , ≠ end_POSTSUPERSCRIPT. In this case, αIvn,𝛼superscriptsubscript𝐼superscript𝑣𝑛\alpha\in I_{v^{\prime}}^{n,\neq}italic_α ∈ italic_I start_POSTSUBSCRIPT italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n , ≠ end_POSTSUPERSCRIPT, u=sαv𝑢subscript𝑠𝛼𝑣u=s_{\alpha}\!\cdot\!vitalic_u = italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ⋅ italic_v, and u=sαvsuperscript𝑢subscript𝑠𝛼superscript𝑣u^{\prime}=s_{\alpha}\!\cdot\!v^{\prime}italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ⋅ italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT. By Lemma 6.13, ϕ(u)=ϕ(v)=ϕ(v)=ϕ(u)italic-ϕ𝑢italic-ϕ𝑣italic-ϕsuperscript𝑣italic-ϕsuperscript𝑢\phi(u)=\phi(v)=\phi(v^{\prime})=\phi(u^{\prime})italic_ϕ ( italic_u ) = italic_ϕ ( italic_v ) = italic_ϕ ( italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) = italic_ϕ ( italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ). By the induction assumption, u=u𝑢superscript𝑢u=u^{\prime}italic_u = italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT. Thus v=sαu=sαu=v𝑣subscript𝑠𝛼𝑢subscript𝑠𝛼superscript𝑢superscript𝑣v=s_{\alpha}\!\cdot\!u=s_{\alpha}\!\cdot\!u^{\prime}=v^{\prime}italic_v = italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ⋅ italic_u = italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ⋅ italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT.

This finishes the proof of Theorem 2.2.

7.4. Local analysis on Yv0(v)subscript𝑌subscript𝑣0𝑣Y_{v_{0}}(v)italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ), Part II

The following Lemma 7.5 strengthens Lemma 7.4 and completes the local analysis on the sets Yv0(v)subscript𝑌subscript𝑣0𝑣Y_{v_{0}}(v)italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ).

Lemma 7.5.

The element u𝑢uitalic_u in Lemma 7.4 is unique. If 2) of Lemma 7.4 occurs and if αIun,𝛼superscriptsubscript𝐼𝑢𝑛\alpha\in I_{u}^{n,\neq}italic_α ∈ italic_I start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n , ≠ end_POSTSUPERSCRIPT, then yYv0(sαu)𝑦subscript𝑌subscript𝑣0normal-⋅subscript𝑠𝛼𝑢y\in Y_{v_{0}}(s_{\alpha}\!\cdot\!u)italic_y ∈ italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ⋅ italic_u ). If 3) of Lemma 7.4 occurs, then yYv0(m(sα)v)𝑦subscript𝑌subscript𝑣0normal-⋅𝑚subscript𝑠𝛼𝑣y\in Y_{v_{0}}(m(s_{\alpha})\!\cdot\!v)italic_y ∈ italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_m ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ) ⋅ italic_v ).

Proof.

The only case where u𝑢uitalic_u might not be unique is when 2) in Lemma 7.4 occurs and when αIun,𝛼superscriptsubscript𝐼𝑢𝑛\alpha\in I_{u}^{n,\neq}italic_α ∈ italic_I start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n , ≠ end_POSTSUPERSCRIPT. Assume this is the case. Since ϕ(u)=ϕ(sαu)italic-ϕ𝑢italic-ϕsubscript𝑠𝛼𝑢\phi(u)=\phi(s_{\alpha}\!\cdot\!u)italic_ϕ ( italic_u ) = italic_ϕ ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ⋅ italic_u ), by Theorem 2.2, ysuperscript𝑦y^{\prime}italic_y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT can not be in both Yv0(u)subscript𝑌subscript𝑣0𝑢Y_{v_{0}}(u)italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_u ) and Yv0(sαu)subscript𝑌subscript𝑣0subscript𝑠𝛼𝑢Y_{v_{0}}(s_{\alpha}\!\cdot\!u)italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ⋅ italic_u ), so the choice of u𝑢uitalic_u is unique. Moreover, by 1) of Lemma 7.3, yWv0(sαu)𝑦subscript𝑊subscript𝑣0subscript𝑠𝛼𝑢y\in W_{v_{0}}(s_{\alpha}\!\cdot\!u)italic_y ∈ italic_W start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ⋅ italic_u ). Let y1Yv0(sαu)subscript𝑦1subscript𝑌subscript𝑣0subscript𝑠𝛼𝑢y_{1}\in Y_{v_{0}}(s_{\alpha}\!\cdot\!u)italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∈ italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ⋅ italic_u ) be such that y1ysubscript𝑦1𝑦y_{1}\leq yitalic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≤ italic_y. By 2) of Lemma 7.3, sαy1Wv0(v)subscript𝑠𝛼subscript𝑦1subscript𝑊subscript𝑣0𝑣s_{\alpha}\ast y_{1}\in W_{v_{0}}(v)italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ∗ italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∈ italic_W start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ). Since sαy1sαy=ysubscript𝑠𝛼subscript𝑦1subscript𝑠𝛼𝑦𝑦s_{\alpha}\ast y_{1}\leq s_{\alpha}\ast y=yitalic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ∗ italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≤ italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ∗ italic_y = italic_y, one has sαy1=ysubscript𝑠𝛼subscript𝑦1𝑦s_{\alpha}\ast y_{1}=yitalic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ∗ italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_y. If sαy1>y1subscript𝑠𝛼subscript𝑦1subscript𝑦1s_{\alpha}y_{1}>y_{1}italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT > italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, then sαy1=sαy1=ysubscript𝑠𝛼subscript𝑦1subscript𝑠𝛼subscript𝑦1𝑦s_{\alpha}y_{1}=s_{\alpha}\ast y_{1}=yitalic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ∗ italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_y, so y1=sαy=ysubscript𝑦1subscript𝑠𝛼𝑦superscript𝑦y_{1}=s_{\alpha}y=y^{\prime}italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT italic_y = italic_y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT, contradicting the fact that ysuperscript𝑦y^{\prime}italic_y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT can not be in both Yv0(u)subscript𝑌subscript𝑣0𝑢Y_{v_{0}}(u)italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_u ) and Yv0(sαu)subscript𝑌subscript𝑣0subscript𝑠𝛼𝑢Y_{v_{0}}(s_{\alpha}\!\cdot\!u)italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ⋅ italic_u ). Thus sαy1<y1subscript𝑠𝛼subscript𝑦1subscript𝑦1s_{\alpha}y_{1}<y_{1}italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT < italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, and hence y=sαy1=y1Yv0(sαu)𝑦subscript𝑠𝛼subscript𝑦1subscript𝑦1subscript𝑌subscript𝑣0subscript𝑠𝛼𝑢y=s_{\alpha}\ast y_{1}=y_{1}\in Y_{v_{0}}(s_{\alpha}\!\cdot\!u)italic_y = italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ∗ italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∈ italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ⋅ italic_u ).

Assume now that 3) of Lemma 7.4 occurs. Then y=sαyWv0(m(sα)v)𝑦subscript𝑠𝛼superscript𝑦subscript𝑊subscript𝑣0𝑚subscript𝑠𝛼𝑣y=s_{\alpha}y^{\prime}\in W_{v_{0}}(m(s_{\alpha})\!\cdot\!v)italic_y = italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT italic_y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_W start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_m ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ) ⋅ italic_v ) by 2) of Lemma 7.3. Let y2Yv0(m(sα)v)subscript𝑦2subscript𝑌subscript𝑣0𝑚subscript𝑠𝛼𝑣y_{2}\in Y_{v_{0}}(m(s_{\alpha})\!\cdot\!v)italic_y start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∈ italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_m ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ) ⋅ italic_v ) be such that y2ysubscript𝑦2𝑦y_{2}\leq yitalic_y start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ≤ italic_y. By 1) of Lemma 7.3, y2Wv0(v)subscript𝑦2subscript𝑊subscript𝑣0𝑣y_{2}\in W_{v_{0}}(v)italic_y start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∈ italic_W start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ). Thus y=y2Yv0(m(sα)v)𝑦subscript𝑦2subscript𝑌subscript𝑣0𝑚subscript𝑠𝛼𝑣y=y_{2}\in Y_{v_{0}}(m(s_{\alpha})\!\cdot\!v)italic_y = italic_y start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∈ italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_m ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ) ⋅ italic_v ).

Q.E.D.

7.5. Elements of Yv0(v)subscript𝑌subscript𝑣0𝑣Y_{v_{0}}(v)italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) and subexpressions

We now prove the following key property of yYv0(v)𝑦subscript𝑌subscript𝑣0𝑣y\in Y_{v_{0}}(v)italic_y ∈ italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) in terms of subexpressions of reduced decompositions of m(y)v0𝑚𝑦subscript𝑣0m(y)\!\cdot\!v_{0}italic_m ( italic_y ) ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT.

Proposition 7.6.

Let v0V0,vVformulae-sequencesubscript𝑣0subscript𝑉0𝑣𝑉v_{0}\in V_{0},v\in Vitalic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v ∈ italic_V and yYv0(v)𝑦subscript𝑌subscript𝑣0𝑣y\in Y_{v_{0}}(v)italic_y ∈ italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ). Then for any reduced word 𝐬=(sα1,,sαk)𝐬subscript𝑠subscript𝛼1normal-…subscript𝑠subscript𝛼𝑘{\bf s}=(s_{\alpha_{1}},\ldots,s_{\alpha_{k}})bold_s = ( italic_s start_POSTSUBSCRIPT italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT , … , italic_s start_POSTSUBSCRIPT italic_α start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) of y-1superscript𝑦1y^{-1}italic_y start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT, there is exactly one subexpression 𝐮=(v0,u1,,uk)𝐮subscript𝑣0subscript𝑢1normal-…subscript𝑢𝑘{\bf u}=(v_{0},u_{1},\ldots,u_{k})bold_u = ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_u start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_u start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) of the reduced decomposition (𝐯v0(𝐬),𝐬)subscript𝐯subscript𝑣0𝐬𝐬({\bf v}_{v_{0}}({\bf s}),{\bf s})( bold_v start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( bold_s ) , bold_s ) of m(y)v0normal-⋅𝑚𝑦subscript𝑣0m(y)\!\cdot\!v_{0}italic_m ( italic_y ) ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT that has final term v𝑣vitalic_v. Moreover, for any 1jk1𝑗𝑘1\leq j\leq k1 ≤ italic_j ≤ italic_k, ujuj-1subscript𝑢𝑗subscript𝑢𝑗1u_{j}\neq u_{j-1}italic_u start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ≠ italic_u start_POSTSUBSCRIPT italic_j - 1 end_POSTSUBSCRIPT and sαjsαj-1sα1Yv0(uj)subscript𝑠subscript𝛼𝑗subscript𝑠subscript𝛼𝑗1normal-⋯subscript𝑠subscript𝛼1subscript𝑌subscript𝑣0subscript𝑢𝑗s_{\alpha_{j}}s_{\alpha_{j-1}}\cdots s_{\alpha_{1}}\in Y_{v_{0}}(u_{j})italic_s start_POSTSUBSCRIPT italic_α start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_s start_POSTSUBSCRIPT italic_α start_POSTSUBSCRIPT italic_j - 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⋯ italic_s start_POSTSUBSCRIPT italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∈ italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_u start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ).

Proof.

By Proposition 6.16, there is a subexpression 𝐮=(u0=v0,u1,,uk-1,uk)fragmentsufragments(subscript𝑢0subscript𝑣0,subscript𝑢1,,subscript𝑢𝑘1,subscript𝑢𝑘){\bf u}=(u_{0}=v_{0},u_{1},\ldots,u_{k-1},u_{k})bold_u = ( italic_u start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_u start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_u start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT , italic_u start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) of the reduced decomposition (𝐯v0(𝐬),𝐬)subscript𝐯subscript𝑣0𝐬𝐬({\bf v}_{v_{0}}({\bf s}),{\bf s})( bold_v start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( bold_s ) , bold_s ) of m(y)v0𝑚𝑦subscript𝑣0m(y)\!\cdot\!v_{0}italic_m ( italic_y ) ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT with final term v𝑣vitalic_v. Letting α=αk𝛼subscript𝛼𝑘\alpha=\alpha_{k}italic_α = italic_α start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT and y=sαk-1sα1superscript𝑦subscript𝑠subscript𝛼𝑘1subscript𝑠subscript𝛼1y^{\prime}=s_{\alpha_{k-1}}\cdots s_{\alpha_{1}}italic_y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_s start_POSTSUBSCRIPT italic_α start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⋯ italic_s start_POSTSUBSCRIPT italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT, one knows from Lemma 7.4 and Lemma 7.5 that yYv0(uk-1)superscript𝑦subscript𝑌subscript𝑣0subscript𝑢𝑘1y^{\prime}\in Y_{v_{0}}(u_{k-1})italic_y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_u start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT ) and that uk-1subscript𝑢𝑘1u_{k-1}italic_u start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT is uniquely determined by the quadruple (v0,v,y,α)subscript𝑣0𝑣𝑦𝛼(v_{0},v,y,\alpha)( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v , italic_y , italic_α ). Proceeding inductively, one proves Proposition 7.6.

Q.E.D.

Notation 7.7.

For v0V0subscript𝑣0subscript𝑉0v_{0}\in V_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, vV𝑣𝑉v\in Vitalic_v ∈ italic_V, yYv0(v)𝑦subscript𝑌subscript𝑣0𝑣y\in Y_{v_{0}}(v)italic_y ∈ italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ), and 𝐬R(y-1)𝐬𝑅superscript𝑦1{\bf s}\in R(y^{-1})bold_s ∈ italic_R ( italic_y start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ), the unique subexpression of the reduced decomposition (𝐯v0(𝐬),𝐬)subscript𝐯subscript𝑣0𝐬𝐬({\bf v}_{v_{0}}({\bf s}),{\bf s})( bold_v start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( bold_s ) , bold_s ) of m(y)v0𝑚𝑦subscript𝑣0m(y)\!\cdot\!v_{0}italic_m ( italic_y ) ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT (see Lemma-Notation 7.2) with final term v𝑣vitalic_v will be denoted by 𝐮v0,v(𝐬)subscript𝐮subscript𝑣0𝑣𝐬{\bf u}_{v_{\scriptscriptstyle 0},v}({\bf s})bold_u start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v end_POSTSUBSCRIPT ( bold_s ).

8. Admissible paths and the sets Yv0(v)subscript𝑌subscript𝑣0𝑣Y_{v_{0}}(v)italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) and Zv0(v)subscript𝑍subscript𝑣0𝑣Z_{v_{0}}(v)italic_Z start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v )

8.1. Admissible paths

Proposition 7.6 leads naturally to the following notion of admissible paths.

Definition 8.1.

Fix v0V0subscript𝑣0subscript𝑉0v_{0}\in V_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT.

i) For vV𝑣𝑉v\in Vitalic_v ∈ italic_V, an admissible path from v0subscript𝑣0v_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT to v𝑣vitalic_v is a pair (𝐯,𝐬)𝐯𝐬({\bf v},{\bf s})( bold_v , bold_s ), where 𝐯=(v0,v1,,vk=v)fragmentsvfragments(subscript𝑣0,subscript𝑣1,,subscript𝑣𝑘v){\bf v}=(v_{0},v_{1},\ldots,v_{k}=v)bold_v = ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_v start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = italic_v ) is a sequence in V𝑉Vitalic_V, and 𝐬=(sα1,sα2,,sαk)𝐬subscript𝑠subscript𝛼1subscript𝑠subscript𝛼2subscript𝑠subscript𝛼𝑘{\bf s}=(s_{\alpha_{1}},s_{\alpha_{2}},\ldots,s_{\alpha_{k}})bold_s = ( italic_s start_POSTSUBSCRIPT italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT , italic_s start_POSTSUBSCRIPT italic_α start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT , … , italic_s start_POSTSUBSCRIPT italic_α start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) is a sequence of simple reflections in W𝑊Witalic_W, and k1𝑘1k\geq 1italic_k ≥ 1, such that one of the following holds for each j[1,k]𝑗1𝑘j\in[1,k]italic_j ∈ [ 1 , italic_k ]:

1) αjCvj-1+Ivj-1nsubscript𝛼𝑗superscriptsubscript𝐶subscript𝑣𝑗1superscriptsubscript𝐼subscript𝑣𝑗1𝑛\alpha_{j}\in C_{v_{j-1}}^{+}\cup I_{v_{j-1}}^{n}italic_α start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∈ italic_C start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT italic_j - 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ∪ italic_I start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT italic_j - 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT and vj=m(sαj)vj-1subscript𝑣𝑗𝑚subscript𝑠subscript𝛼𝑗subscript𝑣𝑗1v_{j}=m(s_{\alpha_{j}})\!\cdot\!v_{j-1}italic_v start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = italic_m ( italic_s start_POSTSUBSCRIPT italic_α start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) ⋅ italic_v start_POSTSUBSCRIPT italic_j - 1 end_POSTSUBSCRIPT;

2) αjIvj-1n,subscript𝛼𝑗superscriptsubscript𝐼subscript𝑣𝑗1𝑛\alpha_{j}\in I_{v_{j-1}}^{n,\neq}italic_α start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∈ italic_I start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT italic_j - 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n , ≠ end_POSTSUPERSCRIPT and vj=sαjvj-1subscript𝑣𝑗subscript𝑠subscript𝛼𝑗subscript𝑣𝑗1v_{j}=s_{\alpha_{j}}\!\cdot\!v_{j-1}italic_v start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = italic_s start_POSTSUBSCRIPT italic_α start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⋅ italic_v start_POSTSUBSCRIPT italic_j - 1 end_POSTSUBSCRIPT.

The pair (𝐯={v0},𝐬=)formulae-sequence𝐯subscript𝑣0𝐬({\bf v}=\{v_{0}\},\,{\bf s}=\emptyset)( bold_v = { italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT } , bold_s = ∅ ) is called the trivial admissible path from v0subscript𝑣0v_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT to v0subscript𝑣0v_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT with the understanding that k=0𝑘0k=0italic_k = 0 in this case. The set of all admissible paths from v0subscript𝑣0v_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT to v𝑣vitalic_v is denoted by 𝒫(v0,v)𝒫subscript𝑣0𝑣{\mathcal{P}}(v_{0},v)caligraphic_P ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v ).

ii) For (𝐯,𝐬)𝒫(v0,v)𝐯𝐬𝒫subscript𝑣0𝑣({\bf v},{\bf s})\in{\mathcal{P}}(v_{0},v)( bold_v , bold_s ) ∈ caligraphic_P ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v ) as in i), the number k𝑘kitalic_k is called the length of (𝐯,𝐬)𝐯𝐬({\bf v},{\bf s})( bold_v , bold_s ) and denoted by l(𝐯,𝐬)𝑙𝐯𝐬l({\bf v},{\bf s})italic_l ( bold_v , bold_s ). We also set y0(𝐯,𝐬)=1Wsubscript𝑦0𝐯𝐬1𝑊y_{0}({\bf v},{\bf s})=1\in Witalic_y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( bold_v , bold_s ) = 1 ∈ italic_W, yj(𝐯,𝐬)=sαjsαj-1sα1subscript𝑦𝑗𝐯𝐬subscript𝑠subscript𝛼𝑗subscript𝑠subscript𝛼𝑗1subscript𝑠subscript𝛼1y_{j}({\bf v},{\bf s})=s_{\alpha_{j}}s_{\alpha_{j-1}}\cdots s_{\alpha_{1}}italic_y start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( bold_v , bold_s ) = italic_s start_POSTSUBSCRIPT italic_α start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_s start_POSTSUBSCRIPT italic_α start_POSTSUBSCRIPT italic_j - 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⋯ italic_s start_POSTSUBSCRIPT italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT for j[1,k]𝑗1𝑘j\in[1,k]italic_j ∈ [ 1 , italic_k ], and y(𝐯,𝐬)=yk(𝐯,𝐬)𝑦𝐯𝐬subscript𝑦𝑘𝐯𝐬y({\bf v},{\bf s})=y_{k}({\bf v},{\bf s})italic_y ( bold_v , bold_s ) = italic_y start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( bold_v , bold_s ).

iii) For vV𝑣𝑉v\in Vitalic_v ∈ italic_V, (𝐯,𝐬)𝒫(v0,v)𝐯𝐬𝒫subscript𝑣0𝑣({\bf v},{\bf s})\in{\mathcal{P}}(v_{0},v)( bold_v , bold_s ) ∈ caligraphic_P ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v ) is said to be a shortest admissible path from v0subscript𝑣0v_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT to v𝑣vitalic_v if l(𝐯,𝐬)l(𝐯,𝐬)𝑙𝐯𝐬𝑙superscript𝐯superscript𝐬l({\bf v},{\bf s})\leq l({\bf v}^{\prime},{\bf s}^{\prime})italic_l ( bold_v , bold_s ) ≤ italic_l ( bold_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , bold_s start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) for every (𝐯,𝐬)𝒫(v0,v)superscript𝐯superscript𝐬𝒫subscript𝑣0𝑣({\bf v}^{\prime},{\bf s}^{\prime})\in{{\mathcal{P}}(v_{0},v)}( bold_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , bold_s start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ∈ caligraphic_P ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v ). The length of a shortest admissible path from v0subscript𝑣0v_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT to v𝑣vitalic_v will be denoted by lv0(v)subscript𝑙subscript𝑣0𝑣l_{v_{0}}(v)italic_l start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ).

Remark 8.2.

We are using the symbol (𝐯,𝐬)𝐯𝐬({\bf v},{\bf s})( bold_v , bold_s ) to denote both a reduced decomposition of an element in V𝑉Vitalic_V and an admissible path in V𝑉Vitalic_V. This should cause no confusion as we will always use the modifiers “reduced decomposition” or “admissible path” in front of (𝐯,𝐬)𝐯𝐬({\bf v},{\bf s})( bold_v , bold_s ). Moreover, it is clear from the definitions that a reduced decomposition of vV𝑣𝑉v\in Vitalic_v ∈ italic_V starting from v0V0subscript𝑣0subscript𝑉0v_{0}\in V_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT is an admissible path from v0subscript𝑣0v_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT to v𝑣vitalic_v.

Lemma 8.3.

Let v0V0subscript𝑣0subscript𝑉0v_{0}\in V_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and vV𝑣𝑉v\in Vitalic_v ∈ italic_V. Let (𝐯,𝐬)𝒫(v0,v)𝐯𝐬𝒫subscript𝑣0𝑣({\bf v},{\bf s})\in{{\mathcal{P}}(v_{0},v)}( bold_v , bold_s ) ∈ caligraphic_P ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v ), where 𝐬=(s1,sk)𝐬subscript𝑠1normal-…subscript𝑠𝑘{\bf s}=(s_{1},\dots s_{k})bold_s = ( italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … italic_s start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ). Then vm(sks1)v0𝑣normal-⋅𝑚normal-∗subscript𝑠𝑘normal-⋯subscript𝑠1subscript𝑣0v\leq m(s_{k}\ast\cdots\ast s_{1})\!\cdot\!v_{0}italic_v ≤ italic_m ( italic_s start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∗ ⋯ ∗ italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT.

Proof.

Let 𝐯=(v0,v1,vk-1,vk=v)fragmentsvfragments(subscript𝑣0,subscript𝑣1,subscript𝑣𝑘1,subscript𝑣𝑘v){\bf v}=(v_{0},v_{1}\ldots,v_{k-1},v_{k}=v)bold_v = ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT … , italic_v start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = italic_v ). Then

vm(sk)vk-1m(sk)m(sk-1)m(s1)v0=m(sks1)v0.𝑣𝑚subscript𝑠𝑘subscript𝑣𝑘1𝑚subscript𝑠𝑘𝑚subscript𝑠𝑘1𝑚subscript𝑠1subscript𝑣0𝑚subscript𝑠𝑘subscript𝑠1subscript𝑣0v\leq m(s_{k})\!\cdot\!v_{{k-1}}\leq\cdots\leq m(s_{k})m(s_{{k-1}})\cdots m(s_% {1})\!\cdot\!v_{0}=m(s_{k}\ast\cdots\ast s_{1})\!\cdot\!v_{0}.italic_v ≤ italic_m ( italic_s start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ⋅ italic_v start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT ≤ ⋯ ≤ italic_m ( italic_s start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) italic_m ( italic_s start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT ) ⋯ italic_m ( italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_m ( italic_s start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∗ ⋯ ∗ italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT .

Q.E.D.

8.2. Minimal admissible paths and elements in Yv0(v)subscript𝑌subscript𝑣0𝑣Y_{v_{0}}(v)italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v )

By Definition 6.15, for v0V0,vV,yYv0(v)formulae-sequencesubscript𝑣0subscript𝑉0formulae-sequence𝑣𝑉𝑦subscript𝑌subscript𝑣0𝑣v_{0}\in V_{0},v\in V,y\in Y_{v_{0}}(v)italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v ∈ italic_V , italic_y ∈ italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ), and 𝐬R(y-1)𝐬𝑅superscript𝑦1{\bf s}\in R(y^{-1})bold_s ∈ italic_R ( italic_y start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ), the pair (𝐮v0,v(𝐬),𝐬)subscript𝐮subscript𝑣0𝑣𝐬𝐬({\bf u}_{v_{0},v}({\bf s}),{\bf s})( bold_u start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v end_POSTSUBSCRIPT ( bold_s ) , bold_s ) (see Notation 7.7) is an admissible path from v0subscript𝑣0v_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT to v𝑣vitalic_v. In this section, we give a characterization of such admissible paths.

Definition 8.4.

Let v0V0subscript𝑣0subscript𝑉0v_{0}\in V_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and vV𝑣𝑉v\in Vitalic_v ∈ italic_V. For a sequence 𝐬𝐬{\bf s}bold_s of simple reflections in W𝑊Witalic_W, let

𝒫(v0,v,𝐬)={(𝐯,𝐬)𝒫(v0,v):𝐬is a subsequence of𝐬}.𝒫subscript𝑣0𝑣𝐬conditional-setsuperscript𝐯superscript𝐬𝒫subscript𝑣0𝑣superscript𝐬is a subsequence of𝐬{\mathcal{P}}(v_{0},v,{\bf s})=\{({\bf v}^{\prime},{\bf s}^{\prime})\in{% \mathcal{P}}(v_{0},v):\;{\bf s}^{\prime}\;\mbox{is a subsequence of}\;{\bf s}\}.caligraphic_P ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v , bold_s ) = { ( bold_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , bold_s start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ∈ caligraphic_P ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v ) : bold_s start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT is a subsequence of bold_s } .

A path (𝐯,𝐬)𝒫(v0,v)𝐯𝐬𝒫subscript𝑣0𝑣({\bf v},{\bf s})\in{\mathcal{P}}(v_{0},v)( bold_v , bold_s ) ∈ caligraphic_P ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v ) is said to be minimal if (𝐯,𝐬)𝐯𝐬({\bf v},{\bf s})( bold_v , bold_s ) is the only member of 𝒫(v0,v,𝐬)𝒫subscript𝑣0𝑣𝐬{\mathcal{P}}(v_{0},v,{\bf s})caligraphic_P ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v , bold_s ). The set of all minimal admissible paths from v0subscript𝑣0v_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT to v𝑣vitalic_v will be denoted by 𝒫min(v0,v)subscript𝒫minsubscript𝑣0𝑣{\mathcal{P}}_{\rm min}(v_{0},v)caligraphic_P start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v ).

Lemma 8.5.

Let v0V0subscript𝑣0subscript𝑉0v_{0}\in V_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and vV𝑣𝑉v\in Vitalic_v ∈ italic_V.

1) If yYv0(v)𝑦subscript𝑌subscript𝑣0𝑣y\in Y_{v_{0}}(v)italic_y ∈ italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) and 𝐬R(y-1)𝐬𝑅superscript𝑦1{\bf s}\in R(y^{-1})bold_s ∈ italic_R ( italic_y start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ), then (𝐮v0,v(𝐬),𝐬)𝒫min(v0,v)subscript𝐮subscript𝑣0𝑣𝐬𝐬subscript𝒫normal-minsubscript𝑣0𝑣({\bf u}_{v_{0},v}({\bf s}),{\bf s})\in{\mathcal{P}}_{\rm min}(v_{0},v)( bold_u start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v end_POSTSUBSCRIPT ( bold_s ) , bold_s ) ∈ caligraphic_P start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v );

2) If (𝐯,𝐬)𝒫min(v0,v)𝐯𝐬subscript𝒫normal-minsubscript𝑣0𝑣({\bf v},{\bf s})\in{\mathcal{P}}_{\rm min}(v_{0},v)( bold_v , bold_s ) ∈ caligraphic_P start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v ) and y=y(𝐯,𝐬)𝑦𝑦𝐯𝐬y=y({\bf v},{\bf s})italic_y = italic_y ( bold_v , bold_s ), then yYv0(v)𝑦subscript𝑌subscript𝑣0𝑣y\in Y_{v_{0}}(v)italic_y ∈ italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ), 𝐬R(y-1)𝐬𝑅superscript𝑦1{\bf s}\in R(y^{-1})bold_s ∈ italic_R ( italic_y start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ), and (𝐯,𝐬)=(𝐮v0,v(𝐬),𝐬)𝐯𝐬subscript𝐮subscript𝑣0𝑣𝐬𝐬({\bf v},{\bf s})=({\bf u}_{v_{0},v}({\bf s}),{\bf s})( bold_v , bold_s ) = ( bold_u start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v end_POSTSUBSCRIPT ( bold_s ) , bold_s ).

Proof.

1) Let yYv0(v)𝑦subscript𝑌subscript𝑣0𝑣y\in Y_{v_{0}}(v)italic_y ∈ italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) and 𝐬R(y-1)𝐬𝑅superscript𝑦1{\bf s}\in R(y^{-1})bold_s ∈ italic_R ( italic_y start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ). Write 𝐬=(s1,,sk)𝐬subscript𝑠1subscript𝑠𝑘{\bf s}=(s_{1},\ldots,s_{k})bold_s = ( italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_s start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ), and let 𝐬=(si1,,sip)superscript𝐬subscript𝑠subscript𝑖1subscript𝑠subscript𝑖𝑝{\bf s}^{\prime}=(s_{i_{1}},\ldots,s_{i_{p}})bold_s start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = ( italic_s start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT , … , italic_s start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) be a subsequence of 𝐬𝐬{\bf s}bold_s. Suppose that (𝐯,𝐬)𝒫(v0,v)superscript𝐯superscript𝐬𝒫subscript𝑣0𝑣({\bf v}^{\prime},{\bf s}^{\prime})\in{{\mathcal{P}}(v_{0},v)}( bold_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , bold_s start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ∈ caligraphic_P ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v ). By Lemma 8.3, vm(sipsi1)v0𝑣𝑚subscript𝑠subscript𝑖𝑝subscript𝑠subscript𝑖1subscript𝑣0v\leq m(s_{i_{p}}\ast\cdots\ast s_{i_{1}})\!\cdot\!v_{0}italic_v ≤ italic_m ( italic_s start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∗ ⋯ ∗ italic_s start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT. Since sipsi1ysubscript𝑠subscript𝑖𝑝subscript𝑠subscript𝑖1𝑦s_{i_{p}}\ast\cdots\ast s_{i_{1}}\leq yitalic_s start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∗ ⋯ ∗ italic_s start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ≤ italic_y, one must have sipsi1=ysubscript𝑠subscript𝑖𝑝subscript𝑠subscript𝑖1𝑦s_{i_{p}}\ast\cdots\ast s_{i_{1}}=yitalic_s start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∗ ⋯ ∗ italic_s start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT = italic_y, which is possible only when 𝐬=𝐬superscript𝐬𝐬{\bf s}^{\prime}={\bf s}bold_s start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = bold_s. It follows that 𝐯superscript𝐯{\bf v}^{\prime}bold_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT is a subexpression of the reduced decomposition (𝐯v0(𝐬),𝐬)subscript𝐯subscript𝑣0𝐬𝐬({\bf v}_{v_{0}}({\bf s}),{\bf s})( bold_v start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( bold_s ) , bold_s ) of m(y)v0𝑚𝑦subscript𝑣0m(y)\!\cdot\!v_{0}italic_m ( italic_y ) ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT. By Proposition 7.6, 𝐯=𝐮v0,v(𝐬)superscript𝐯subscript𝐮subscript𝑣0𝑣𝐬{\bf v}^{\prime}={\bf u}_{v_{0},v}({\bf s})bold_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = bold_u start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v end_POSTSUBSCRIPT ( bold_s ). This proves that (𝐮v0,v(𝐬),𝐬)𝒫min(v0,v)subscript𝐮subscript𝑣0𝑣𝐬𝐬subscript𝒫minsubscript𝑣0𝑣({\bf u}_{v_{0},v}({\bf s}),{\bf s})\in{\mathcal{P}}_{\rm min}(v_{0},v)( bold_u start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v end_POSTSUBSCRIPT ( bold_s ) , bold_s ) ∈ caligraphic_P start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v ).

2) Let (𝐯,𝐬)𝒫min(v0,v)𝐯𝐬subscript𝒫minsubscript𝑣0𝑣({\bf v},{\bf s})\in{\mathcal{P}}_{\rm min}(v_{0},v)( bold_v , bold_s ) ∈ caligraphic_P start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v ) and let y=y(𝐯,𝐬)𝑦𝑦𝐯𝐬y=y({\bf v},{\bf s})italic_y = italic_y ( bold_v , bold_s ). Let 𝐬=(s1,,sk)𝐬subscript𝑠1subscript𝑠𝑘{\bf s}=(s_{1},\ldots,s_{k})bold_s = ( italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_s start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ). By Lemma 8.3, vm(sks1)v0𝑣𝑚subscript𝑠𝑘subscript𝑠1subscript𝑣0v\leq m(s_{k}\ast\cdots\ast s_{1})\!\cdot\!v_{0}italic_v ≤ italic_m ( italic_s start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∗ ⋯ ∗ italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT. Suppose that sks1sks1subscript𝑠𝑘subscript𝑠1subscript𝑠𝑘subscript𝑠1s_{k}\ast\cdots\ast s_{1}\neq s_{k}\cdots s_{1}italic_s start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∗ ⋯ ∗ italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≠ italic_s start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ⋯ italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT. Then there exists a proper subsequence 𝐬=(si1,,sip)superscript𝐬subscript𝑠subscript𝑖1subscript𝑠subscript𝑖𝑝{\bf s}^{\prime}=(s_{i_{1}},\ldots,s_{i_{p}})bold_s start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = ( italic_s start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT , … , italic_s start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) of 𝐬𝐬{\bf s}bold_s such that 𝐬superscript𝐬{\bf s}^{\prime}bold_s start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT is a reduced word of y=sks1=sipsi1superscript𝑦subscript𝑠𝑘subscript𝑠1subscript𝑠subscript𝑖𝑝subscript𝑠subscript𝑖1y^{\prime}=s_{k}\ast\cdots\ast s_{1}=s_{i_{p}}\cdots s_{i_{1}}italic_y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_s start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∗ ⋯ ∗ italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_s start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⋯ italic_s start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT. By passing to a subsequence of 𝐬superscript𝐬{\bf s}^{\prime}bold_s start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT if necessary, we can assume that yYv0(v)superscript𝑦subscript𝑌subscript𝑣0𝑣y^{\prime}\in Y_{v_{0}}(v)italic_y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ). Then (𝐮v0,v(𝐬),𝐬)subscript𝐮subscript𝑣0𝑣superscript𝐬superscript𝐬({\bf u}_{v_{0},v}({\bf s}^{\prime}),{\bf s}^{\prime})( bold_u start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v end_POSTSUBSCRIPT ( bold_s start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) , bold_s start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) is an admissible path from v0subscript𝑣0v_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT to v𝑣vitalic_v that is different from (𝐯,𝐬)𝐯𝐬({\bf v},{\bf s})( bold_v , bold_s ), contradicting the assumption on (𝐯,𝐬)𝐯𝐬({\bf v},{\bf s})( bold_v , bold_s ). Thus y=sks1=sks1𝑦subscript𝑠𝑘subscript𝑠1subscript𝑠𝑘subscript𝑠1y=s_{k}\ast\cdots\ast s_{1}=s_{k}\cdots s_{1}italic_y = italic_s start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∗ ⋯ ∗ italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_s start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ⋯ italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, so 𝐬𝐬{\bf s}bold_s is a reduced word of y-1superscript𝑦1y^{-1}italic_y start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT. The same arguments show that yYv0(v)𝑦subscript𝑌subscript𝑣0𝑣y\in Y_{v_{0}}(v)italic_y ∈ italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ). By Proposition 7.6, (𝐯,𝐬)=(𝐮v0,v(𝐬),𝐬)𝐯𝐬subscript𝐮subscript𝑣0𝑣𝐬𝐬({\bf v},{\bf s})=({\bf u}_{v_{0},v}({\bf s}),{\bf s})( bold_v , bold_s ) = ( bold_u start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v end_POSTSUBSCRIPT ( bold_s ) , bold_s ).

Q.E.D.

For v0V0subscript𝑣0subscript𝑉0v_{0}\in V_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and vV𝑣𝑉v\in Vitalic_v ∈ italic_V, let

𝒴v0(v)={(y,𝐬):yYv0(v),𝐬R(y-1)}.subscript𝒴subscript𝑣0𝑣conditional-set𝑦𝐬formulae-sequence𝑦subscript𝑌subscript𝑣0𝑣𝐬𝑅superscript𝑦1{\mathcal{Y}}_{v_{0}}(v)=\{(y,\,{\bf s}):\,y\in Y_{v_{0}}(v),\,{\bf s}\in R(y^% {-1})\}.caligraphic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) = { ( italic_y , bold_s ) : italic_y ∈ italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) , bold_s ∈ italic_R ( italic_y start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ) } .

The following Proposition 8.6 follows immediately from Lemma 8.5.

Proposition 8.6.

For any v0V0subscript𝑣0subscript𝑉0v_{0}\in V_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and vV𝑣𝑉v\in Vitalic_v ∈ italic_V, the map

(8.1) Ω:𝒫min(v0,v)𝒴v0(v):(𝐯,𝐬)(y(𝐯,𝐬),𝐬):Ωsubscript𝒫minsubscript𝑣0𝑣subscript𝒴subscript𝑣0𝑣:𝐯𝐬𝑦𝐯𝐬𝐬\Omega:\;\;\;{\mathcal{P}}_{\rm min}(v_{0},v)\longrightarrow{\mathcal{Y}}_{v_{% 0}}(v):\;\;\;({\bf v},{\bf s})\longmapsto(y({\bf v},{\bf s}),\,{\bf s})roman_Ω : caligraphic_P start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v ) ⟶ caligraphic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) : ( bold_v , bold_s ) ⟼ ( italic_y ( bold_v , bold_s ) , bold_s )

is bijective, with inverse given by

(8.2) Ω-1:𝒴v0(v)𝒫min(v0,v):(y,𝐬)(𝐮v0,v(𝐬),𝐬).:superscriptΩ1subscript𝒴subscript𝑣0𝑣subscript𝒫minsubscript𝑣0𝑣:𝑦𝐬subscript𝐮subscript𝑣0𝑣𝐬𝐬\Omega^{-1}:\;\;\;{\mathcal{Y}}_{v_{0}}(v)\longrightarrow{\mathcal{P}}_{\rm min% }(v_{0},v):\;\;\;(y,{\bf s})\longmapsto({\bf u}_{v_{0},v}({\bf s}),{\bf s}).roman_Ω start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT : caligraphic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) ⟶ caligraphic_P start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v ) : ( italic_y , bold_s ) ⟼ ( bold_u start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v end_POSTSUBSCRIPT ( bold_s ) , bold_s ) .
Definition 8.7.

For v0V0,vVformulae-sequencesubscript𝑣0subscript𝑉0𝑣𝑉v_{0}\in V_{0},v\in Vitalic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v ∈ italic_V, and (y,𝐬)𝒴v0(v)𝑦𝐬subscript𝒴subscript𝑣0𝑣(y,{\bf s})\in{\mathcal{Y}}_{v_{0}}(v)( italic_y , bold_s ) ∈ caligraphic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ), we will call (𝐮v0,v(𝐬),𝐬)subscript𝐮subscript𝑣0𝑣𝐬𝐬({\bf u}_{v_{0},v}({\bf s}),{\bf s})( bold_u start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v end_POSTSUBSCRIPT ( bold_s ) , bold_s ) the admissible path from v0subscript𝑣0v_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT to v𝑣vitalic_v associated to 𝐬𝐬{\bf s}bold_s.

Corollary 8.8.

For any v0Vsubscript𝑣0𝑉v_{0}\in Vitalic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ italic_V and vV𝑣𝑉v\in Vitalic_v ∈ italic_V, one has

Yv0(v)={y(𝐯,𝐬):(𝐯,𝐬)𝒫min(v0,v)}.subscript𝑌subscript𝑣0𝑣conditional-set𝑦𝐯𝐬𝐯𝐬subscript𝒫minsubscript𝑣0𝑣Y_{v_{0}}(v)=\{y({\bf v},{\bf s}):({\bf v},{\bf s})\in{\mathcal{P}}_{\rm min}(% v_{0},v)\}.italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) = { italic_y ( bold_v , bold_s ) : ( bold_v , bold_s ) ∈ caligraphic_P start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v ) } .

8.3. Shortest admissible paths and elements in Zv0(v)subscript𝑍subscript𝑣0𝑣Z_{v_{0}}(v)italic_Z start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v )

Let v0V0subscript𝑣0subscript𝑉0v_{0}\in V_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and vV𝑣𝑉v\in Vitalic_v ∈ italic_V. Let 𝒫short(v0,v)subscript𝒫shortsubscript𝑣0𝑣{\mathcal{P}}_{\rm short}(v_{0},v)caligraphic_P start_POSTSUBSCRIPT roman_short end_POSTSUBSCRIPT ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v ) denote the set of all shortest admissible paths from v0subscript𝑣0v_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT to v𝑣vitalic_v (see Definition 8.1). It is clear that 𝒫short(v0,v)𝒫min(v0,v)subscript𝒫shortsubscript𝑣0𝑣subscript𝒫minsubscript𝑣0𝑣{\mathcal{P}}_{\rm short}(v_{0},v)\subset{\mathcal{P}}_{\rm min}(v_{0},v)caligraphic_P start_POSTSUBSCRIPT roman_short end_POSTSUBSCRIPT ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v ) ⊂ caligraphic_P start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v ). Let

𝒵v0(v)={(z,𝐬):zZv0(v),𝐬R(z-1)}.subscript𝒵subscript𝑣0𝑣conditional-set𝑧𝐬formulae-sequence𝑧subscript𝑍subscript𝑣0𝑣𝐬𝑅superscript𝑧1{\mathcal{Z}}_{v_{0}}(v)=\{(z,\,{\bf s}):\,z\in Z_{v_{0}}(v),\,{\bf s}\in R(z^% {-1})\}.caligraphic_Z start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) = { ( italic_z , bold_s ) : italic_z ∈ italic_Z start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) , bold_s ∈ italic_R ( italic_z start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ) } .

Then 𝒵v0(v)𝒴v0(v)subscript𝒵subscript𝑣0𝑣subscript𝒴subscript𝑣0𝑣{\mathcal{Z}}_{v_{0}}(v)\subset{\mathcal{Y}}_{v_{0}}(v)caligraphic_Z start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) ⊂ caligraphic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ).

Proposition 8.9.

For any v0V0subscript𝑣0subscript𝑉0v_{0}\in V_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and vV𝑣𝑉v\in Vitalic_v ∈ italic_V, the map Ωnormal-Ω\Omegaroman_Ω in (8.1) restricts to a bijection between 𝒫short(v0,v)subscript𝒫normal-shortsubscript𝑣0𝑣{\mathcal{P}}_{\rm short}(v_{0},v)caligraphic_P start_POSTSUBSCRIPT roman_short end_POSTSUBSCRIPT ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v ) and 𝒵v0(v)subscript𝒵subscript𝑣0𝑣{\mathcal{Z}}_{v_{0}}(v)caligraphic_Z start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ).

Proof.

Since l(𝐯,𝐬)=l(y(𝐯,𝐬))𝑙𝐯𝐬𝑙𝑦𝐯𝐬l({\bf v},{\bf s})=l(y({\bf v},{\bf s}))italic_l ( bold_v , bold_s ) = italic_l ( italic_y ( bold_v , bold_s ) ) for all (𝐯,𝐬)𝒫min(v0,v)𝐯𝐬subscript𝒫minsubscript𝑣0𝑣({\bf v},{\bf s})\in{\mathcal{P}}_{\rm min}(v_{0},v)( bold_v , bold_s ) ∈ caligraphic_P start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v ), Proposition 8.9 follows directly from Proposition 8.6.

Q.E.D.

Corollary 8.10.

For any v0V0subscript𝑣0subscript𝑉0v_{0}\in V_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and vV𝑣𝑉v\in Vitalic_v ∈ italic_V, one has

Zv0(v)={y(𝐯,𝐬):(𝐯,𝐬)𝒫short(v0,v)}.subscript𝑍subscript𝑣0𝑣conditional-set𝑦𝐯𝐬𝐯𝐬subscript𝒫shortsubscript𝑣0𝑣Z_{v_{0}}(v)=\{y({\bf v},{\bf s}):({\bf v},{\bf s})\in{\mathcal{P}}_{\rm short% }(v_{0},v)\}.italic_Z start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) = { italic_y ( bold_v , bold_s ) : ( bold_v , bold_s ) ∈ caligraphic_P start_POSTSUBSCRIPT roman_short end_POSTSUBSCRIPT ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v ) } .
Example 8.11.

Let v0V0subscript𝑣0subscript𝑉0v_{0}\in V_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and let M(W,S)v0={m(w)v0:wW}𝑀𝑊𝑆subscript𝑣0conditional-set𝑚𝑤subscript𝑣0𝑤𝑊M(W,S)\!\cdot\!v_{0}=\{m(w)\!\cdot\!v_{0}:w\in W\}italic_M ( italic_W , italic_S ) ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = { italic_m ( italic_w ) ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT : italic_w ∈ italic_W }. Suppose that vM(W,S)v0𝑣𝑀𝑊𝑆subscript𝑣0v\in M(W,S)\!\cdot\!v_{0}italic_v ∈ italic_M ( italic_W , italic_S ) ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT. Then

(8.3) Zv0(v)={zW:m(z)v0=v,l(z)=l(v)}.subscript𝑍subscript𝑣0𝑣conditional-set𝑧𝑊formulae-sequence𝑚𝑧subscript𝑣0𝑣𝑙𝑧𝑙𝑣Z_{v_{0}}(v)=\{z\in W:\;m(z)\!\cdot\!v_{0}=v,\,l(z)=l(v)\}.italic_Z start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) = { italic_z ∈ italic_W : italic_m ( italic_z ) ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_v , italic_l ( italic_z ) = italic_l ( italic_v ) } .

Indeed, if wWv0(v)𝑤subscript𝑊subscript𝑣0𝑣w\in W_{v_{0}}(v)italic_w ∈ italic_W start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ), then l(v)l(m(w)v0)l(w).𝑙𝑣𝑙𝑚𝑤subscript𝑣0𝑙𝑤l(v)\leq l(m(w)\!\cdot\!v_{0})\leq l(w).italic_l ( italic_v ) ≤ italic_l ( italic_m ( italic_w ) ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) ≤ italic_l ( italic_w ) . Since vM(W,S)v0𝑣𝑀𝑊𝑆subscript𝑣0v\in M(W,S)\!\cdot\!v_{0}italic_v ∈ italic_M ( italic_W , italic_S ) ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, there exists z0Wsubscript𝑧0𝑊z_{0}\in Witalic_z start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ italic_W such that v=m(z0)v0𝑣𝑚subscript𝑧0subscript𝑣0v=m(z_{0})\!\cdot\!v_{0}italic_v = italic_m ( italic_z start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and l(z0)=l(v)𝑙subscript𝑧0𝑙𝑣l(z_{0})=l(v)italic_l ( italic_z start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) = italic_l ( italic_v ). Thus l(v)=min{l(w):wWv0(v)}𝑙𝑣:𝑙𝑤𝑤subscript𝑊subscript𝑣0𝑣l(v)=\min\{l(w):w\in W_{v_{0}}(v)\}italic_l ( italic_v ) = roman_min { italic_l ( italic_w ) : italic_w ∈ italic_W start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) }. Let Zv0(v)superscriptsubscript𝑍subscript𝑣0𝑣Z_{v_{0}}^{\prime}(v)italic_Z start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_v ) be the set of the right hand side on (8.3). Then Zv0(v)Zv0(v)superscriptsubscript𝑍subscript𝑣0𝑣subscript𝑍subscript𝑣0𝑣Z_{v_{0}}^{\prime}(v)\subset Z_{v_{0}}(v)italic_Z start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_v ) ⊂ italic_Z start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ). Conversely, if zZv0(v)𝑧subscript𝑍subscript𝑣0𝑣z\in Z_{v_{0}}(v)italic_z ∈ italic_Z start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ), then l(z)=l(v)𝑙𝑧𝑙𝑣l(z)=l(v)italic_l ( italic_z ) = italic_l ( italic_v ), and it follows from l(v)l(m(z)v0)l(z)𝑙𝑣𝑙𝑚𝑧subscript𝑣0𝑙𝑧l(v)\leq l(m(z)\!\cdot\!v_{0})\leq l(z)italic_l ( italic_v ) ≤ italic_l ( italic_m ( italic_z ) ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) ≤ italic_l ( italic_z ) that v=m(z)v0𝑣𝑚𝑧subscript𝑣0v=m(z)\!\cdot\!v_{0}italic_v = italic_m ( italic_z ) ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT. Thus Zv0(v)Zv0(v)subscript𝑍subscript𝑣0𝑣superscriptsubscript𝑍subscript𝑣0𝑣Z_{v_{0}}(v)\subset Z_{v_{0}}^{\prime}(v)italic_Z start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) ⊂ italic_Z start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_v ). Hence Zv0(v)=Zv0(v)subscript𝑍subscript𝑣0𝑣superscriptsubscript𝑍subscript𝑣0𝑣Z_{v_{0}}(v)=Z_{v_{0}}^{\prime}(v)italic_Z start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) = italic_Z start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_v ).

We will see in §§\lx@sectionsign§9.4 that it can happen that vM(W,S)v0𝑣𝑀𝑊𝑆subscript𝑣0v\in M(W,S)\!\cdot\!v_{0}italic_v ∈ italic_M ( italic_W , italic_S ) ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and Zv0(v)subscript𝑍subscript𝑣0𝑣Z_{v_{0}}(v)italic_Z start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) is a proper subset of Yv0(v)subscript𝑌subscript𝑣0𝑣Y_{v_{0}}(v)italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ).

9. Examples

9.1. The case when vV0𝑣subscript𝑉0v\in V_{0}italic_v ∈ italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT

In this subsection, we assume that K𝐾Kitalic_K is connected. Fix v0,v0V0subscript𝑣0superscriptsubscript𝑣0subscript𝑉0v_{0},v_{0}^{\prime}\in V_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT. We will consider the set Yv0(v0)subscript𝑌subscript𝑣0superscriptsubscript𝑣0Y_{v_{0}}(v_{0}^{\prime})italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ).

Recall from §§\lx@sectionsign§4.3 that Yv0(v0)=min(Wv0(v0))subscript𝑌subscript𝑣0superscriptsubscript𝑣0superscriptsubscript𝑊subscript𝑣0superscriptsubscript𝑣0Y_{v_{0}}(v_{0}^{\prime})=\min(W_{v_{0}}^{\prime}(v_{0}^{\prime}))italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) = roman_min ( italic_W start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ), where

(9.1) Wv0(v0)={wW:w=p(B,B)for someBK(v0),BK(v0)},superscriptsubscript𝑊subscript𝑣0superscriptsubscript𝑣0conditional-set𝑤𝑊formulae-sequence𝑤𝑝superscript𝐵𝐵for somesuperscript𝐵𝐾superscriptsubscript𝑣0𝐵𝐾subscript𝑣0W_{v_{0}}^{\prime}(v_{0}^{\prime})=\{w\in W:\;w=p(B^{\prime},B)\;\mbox{for % some}\;B^{\prime}\in K(v_{0}^{\prime}),\,B\in K(v_{0})\},italic_W start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) = { italic_w ∈ italic_W : italic_w = italic_p ( italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_B ) for some italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_K ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) , italic_B ∈ italic_K ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) } ,

and p:×W=(×)/G:𝑝𝑊𝐺p:{\mathcal{B}}\times{\mathcal{B}}\to W=({\mathcal{B}}\times{\mathcal{B}})/Gitalic_p : caligraphic_B × caligraphic_B → italic_W = ( caligraphic_B × caligraphic_B ) / italic_G is the natural projection. By Lemma 6.5, Wv0(v0)={wWθ:wv0=v0}.superscriptsubscript𝑊subscript𝑣0superscriptsubscript𝑣0conditional-set𝑤superscript𝑊𝜃𝑤subscript𝑣0superscriptsubscript𝑣0W_{v_{0}}^{\prime}(v_{0}^{\prime})=\{w\in W^{\theta}:w\!\cdot\!v_{0}=v_{0}^{% \prime}\}.italic_W start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) = { italic_w ∈ italic_W start_POSTSUPERSCRIPT italic_θ end_POSTSUPERSCRIPT : italic_w ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT } . Recall from Definition 6.7 the element yv0,v0Wθsubscript𝑦subscript𝑣0superscriptsubscript𝑣0superscript𝑊𝜃y_{v_{0},v_{0}^{\prime}}\in W^{\theta}italic_y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ∈ italic_W start_POSTSUPERSCRIPT italic_θ end_POSTSUPERSCRIPT. By 3) of Proposition 6.8, yv0,v0subscript𝑦subscript𝑣0superscriptsubscript𝑣0y_{v_{0},v_{0}^{\prime}}italic_y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT is the unique minimal element in Wv0(v0)superscriptsubscript𝑊subscript𝑣0superscriptsubscript𝑣0W_{v_{0}}^{\prime}(v_{0}^{\prime})italic_W start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ). Thus Yv0(v0)={yv0,v0}subscript𝑌subscript𝑣0superscriptsubscript𝑣0subscript𝑦subscript𝑣0superscriptsubscript𝑣0Y_{v_{0}}(v_{0}^{\prime})=\{y_{v_{0},v_{0}^{\prime}}\}italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) = { italic_y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT } has one element.

9.2. The case when there is a unique closed orbit

Assume that V0={v0}subscript𝑉0subscript𝑣0V_{0}=\{v_{0}\}italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = { italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT } has only one element. Then M(W,S)v0=V𝑀𝑊𝑆subscript𝑣0𝑉M(W,S)\!\cdot\!v_{0}=Vitalic_M ( italic_W , italic_S ) ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_V. In this case, for every vV𝑣𝑉v\in Vitalic_v ∈ italic_V, one has

Yv0(v)=Zv0(v)={zW:m(z)v0=v,l(z)=l(v)}.subscript𝑌subscript𝑣0𝑣subscript𝑍subscript𝑣0𝑣conditional-set𝑧𝑊formulae-sequence𝑚𝑧subscript𝑣0𝑣𝑙𝑧𝑙𝑣Y_{v_{0}}(v)=Z_{v_{0}}(v)=\{z\in W:\;m(z)\!\cdot\!v_{0}=v,\,l(z)=l(v)\}.italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) = italic_Z start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) = { italic_z ∈ italic_W : italic_m ( italic_z ) ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_v , italic_l ( italic_z ) = italic_l ( italic_v ) } .

Indeed, by [12, Corollary 9.15], the Springer map ϕ:VW:italic-ϕ𝑉𝑊\phi:V\to Witalic_ϕ : italic_V → italic_W is injective. For any vV𝑣𝑉v\in Vitalic_v ∈ italic_V, since ϕ(sαv)=ϕ(v)italic-ϕsubscript𝑠𝛼𝑣italic-ϕ𝑣\phi(s_{\alpha}\!\cdot\!v)=\phi(v)italic_ϕ ( italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ⋅ italic_v ) = italic_ϕ ( italic_v ) for any αIvn𝛼superscriptsubscript𝐼𝑣𝑛\alpha\in I_{v}^{n}italic_α ∈ italic_I start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT, one has Ivn,=superscriptsubscript𝐼𝑣𝑛I_{v}^{n,\neq}=\emptysetitalic_I start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n , ≠ end_POSTSUPERSCRIPT = ∅, and thus for any admissible path (𝐯,𝐬)𝐯𝐬({\bf v},{\bf s})( bold_v , bold_s ) from v0subscript𝑣0v_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT to v𝑣vitalic_v, 2) in Definition 8.1 does not occur. Therefore if yYv0(v)𝑦subscript𝑌subscript𝑣0𝑣y\in Y_{v_{0}}(v)italic_y ∈ italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ), then v=m(y)v0𝑣𝑚𝑦subscript𝑣0v=m(y)\!\cdot\!v_{0}italic_v = italic_m ( italic_y ) ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and l(y)=l(v)𝑙𝑦𝑙𝑣l(y)=l(v)italic_l ( italic_y ) = italic_l ( italic_v ), and so yZv0(v)𝑦subscript𝑍subscript𝑣0𝑣y\in Z_{v_{0}}(v)italic_y ∈ italic_Z start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ).

Remark 9.1.

For any vV𝑣𝑉v\in Vitalic_v ∈ italic_V, and not assuming that V0subscript𝑉0V_{0}italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT has only one element, T. A. Springer has studied in [17] the set v0V0{zW:m(z)v0=v,l(z)=l(v)}subscriptsubscript𝑣0subscript𝑉0conditional-set𝑧𝑊formulae-sequence𝑚𝑧subscript𝑣0𝑣𝑙𝑧𝑙𝑣\cup_{v_{0}\in V_{0}}\{z\in W:m(z)\!\cdot\!v_{0}=v,\,l(z)=l(v)\}∪ start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT { italic_z ∈ italic_W : italic_m ( italic_z ) ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_v , italic_l ( italic_z ) = italic_l ( italic_v ) } as an invariant for v𝑣vitalic_v.

9.3. The Hermitian symmetric case

Assume that G𝐺Gitalic_G is simple and simply connected. By [13, Definition 5.1.1], (G,θ)𝐺𝜃(G,\theta)( italic_G , italic_θ ) is said to be of Hermitian symmetric type if the center of K𝐾Kitalic_K has positive dimension.

Assume that (G,θ)𝐺𝜃(G,\theta)( italic_G , italic_θ ) is of Hermitian symmetric type. By [13, Theorem 5.12], there exists a standard pair (B,H)𝒞𝐵𝐻𝒞(B,H)\in{\mathcal{C}}( italic_B , italic_H ) ∈ caligraphic_C and a parabolic subgroup P𝑃Pitalic_P of G𝐺Gitalic_G containing B𝐵Bitalic_B such that K𝐾Kitalic_K is the unique Levi subgroup of P𝑃Pitalic_P containing H𝐻Hitalic_H. Moreover, there exists α0Γsubscript𝛼0Γ\alpha_{0}\in\Gammaitalic_α start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ roman_Γ such that P𝑃Pitalic_P is of type J=Γ\{α0}𝐽\Γsubscript𝛼0J=\Gamma\backslash\{\alpha_{0}\}italic_J = roman_Γ \ { italic_α start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT }. Let v0Vsubscript𝑣0𝑉v_{0}\in Vitalic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ italic_V such that BK(v0)𝐵𝐾subscript𝑣0B\in K(v_{0})italic_B ∈ italic_K ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ). Then every αJ𝛼𝐽\alpha\in Jitalic_α ∈ italic_J is compact imaginary for v0subscript𝑣0v_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT. We now study the set Yv0(v)subscript𝑌subscript𝑣0𝑣Y_{v_{0}}(v)italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) for every vV𝑣𝑉v\in Vitalic_v ∈ italic_V.

Let WJsubscript𝑊𝐽W_{J}italic_W start_POSTSUBSCRIPT italic_J end_POSTSUBSCRIPT be the subgroup of W𝑊Witalic_W generated by simple reflections corresponding to roots in J𝐽Jitalic_J, and let WJsuperscript𝑊𝐽W^{J}italic_W start_POSTSUPERSCRIPT italic_J end_POSTSUPERSCRIPT be the set of minimal length representatives for W/WJ𝑊subscript𝑊𝐽W/W_{J}italic_W / italic_W start_POSTSUBSCRIPT italic_J end_POSTSUBSCRIPT in W𝑊Witalic_W. Then WJsuperscript𝑊𝐽W^{J}italic_W start_POSTSUPERSCRIPT italic_J end_POSTSUPERSCRIPT parametrizes the set of (B,P)𝐵𝑃(B,P)( italic_B , italic_P )-double cosets in G𝐺Gitalic_G via

WJdBdP:=BηB,H-1(d)PG,containssuperscript𝑊𝐽𝑑𝐵𝑑𝑃assign𝐵superscriptsubscript𝜂𝐵𝐻1𝑑𝑃𝐺W^{J}\ni d\longmapsto BdP:=B\eta_{B,H}^{-1}(d)P\subset G,italic_W start_POSTSUPERSCRIPT italic_J end_POSTSUPERSCRIPT ∋ italic_d ⟼ italic_B italic_d italic_P := italic_B italic_η start_POSTSUBSCRIPT italic_B , italic_H end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_d ) italic_P ⊂ italic_G ,

where ηB,H:WHW:subscript𝜂𝐵𝐻subscript𝑊𝐻𝑊\eta_{B,H}:W_{H}\to Witalic_η start_POSTSUBSCRIPT italic_B , italic_H end_POSTSUBSCRIPT : italic_W start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT → italic_W is given in (3.4). Since every (B,K)𝐵𝐾(B,K)( italic_B , italic_K )-double coset in G𝐺Gitalic_G is contained in a unique (B,P)𝐵𝑃(B,P)( italic_B , italic_P )-double coset, we have the well-defined surjective map

ν:VWJ:ν(v)=dWJ  if  BvKBdP.fragmentsν:Vsuperscript𝑊𝐽:νfragments(v)dsuperscript𝑊𝐽italic-  ifitalic-  BvKBdP.\nu:\;V\longrightarrow W^{J}:\;\;\;\nu(v)=d\in W^{J}\hskip 14.454pt\mbox{{\rm if% }}\hskip 14.454ptBvK\subset BdP.italic_ν : italic_V ⟶ italic_W start_POSTSUPERSCRIPT italic_J end_POSTSUPERSCRIPT : italic_ν ( italic_v ) = italic_d ∈ italic_W start_POSTSUPERSCRIPT italic_J end_POSTSUPERSCRIPT if italic_B italic_v italic_K ⊂ italic_B italic_d italic_P .

It is proved in [13, Theorem 5.2.5] that the map

η:VW×WJ:η(v)=(ϕ(v),ν(v)):𝜂𝑉𝑊superscript𝑊𝐽:𝜂𝑣italic-ϕ𝑣𝜈𝑣\eta:\;V\longrightarrow W\times W^{J}:\;\;\eta(v)=(\phi(v),\;\nu(v))italic_η : italic_V ⟶ italic_W × italic_W start_POSTSUPERSCRIPT italic_J end_POSTSUPERSCRIPT : italic_η ( italic_v ) = ( italic_ϕ ( italic_v ) , italic_ν ( italic_v ) )

is injective.

Proposition 9.2.

Let (G,θ)𝐺𝜃(G,\theta)( italic_G , italic_θ ) be of Hermitian symmetric type and let the notation be as above. Then for any vV𝑣𝑉v\in Vitalic_v ∈ italic_V, ν(v)WJ𝜈𝑣superscript𝑊𝐽\nu(v)\in W^{J}italic_ν ( italic_v ) ∈ italic_W start_POSTSUPERSCRIPT italic_J end_POSTSUPERSCRIPT is the unique element in Yv0(v)subscript𝑌subscript𝑣0𝑣Y_{v_{0}}(v)italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ).

Proof.

Note that P=BK𝑃𝐵𝐾P=BKitalic_P = italic_B italic_K. Let vV𝑣𝑉v\in Vitalic_v ∈ italic_V. By the definition of ν(v)𝜈𝑣\nu(v)italic_ν ( italic_v ), one has

BvKBν(v)PBν(v)P¯=Bν(v)BK¯=Bν(v)B¯K=B(m(ν(v))v0)K¯.𝐵𝑣𝐾𝐵𝜈𝑣𝑃¯𝐵𝜈𝑣𝑃¯𝐵𝜈𝑣𝐵𝐾¯𝐵𝜈𝑣𝐵𝐾¯𝐵𝑚𝜈𝑣subscript𝑣0𝐾BvK\subset B\nu(v)P\subset\overline{B\nu(v)P}=\overline{B\nu(v)BK}=\overline{B% \nu(v)B}\;K=\overline{B(m(\nu(v))\!\cdot\!v_{0})K}.italic_B italic_v italic_K ⊂ italic_B italic_ν ( italic_v ) italic_P ⊂ ¯ start_ARG italic_B italic_ν ( italic_v ) italic_P end_ARG = ¯ start_ARG italic_B italic_ν ( italic_v ) italic_B italic_K end_ARG = ¯ start_ARG italic_B italic_ν ( italic_v ) italic_B end_ARG italic_K = ¯ start_ARG italic_B ( italic_m ( italic_ν ( italic_v ) ) ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) italic_K end_ARG .

Thus ν(v)Wv0(v)𝜈𝑣subscript𝑊subscript𝑣0𝑣\nu(v)\in W_{v_{0}}(v)italic_ν ( italic_v ) ∈ italic_W start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ). Conversely, assume that wWv0(v)𝑤subscript𝑊subscript𝑣0𝑣w\in W_{v_{0}}(v)italic_w ∈ italic_W start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ). Write w=dx𝑤𝑑𝑥w=dxitalic_w = italic_d italic_x, where dWJ𝑑superscript𝑊𝐽d\in W^{J}italic_d ∈ italic_W start_POSTSUPERSCRIPT italic_J end_POSTSUPERSCRIPT and xWJ𝑥subscript𝑊𝐽x\in W_{J}italic_x ∈ italic_W start_POSTSUBSCRIPT italic_J end_POSTSUBSCRIPT. Since every αJ𝛼𝐽\alpha\in Jitalic_α ∈ italic_J is compact imaginary for v0subscript𝑣0v_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, m(x)v0=v0𝑚𝑥subscript𝑣0subscript𝑣0m(x)\!\cdot\!v_{0}=v_{0}italic_m ( italic_x ) ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT. Thus vm(w)v0=m(d)v0𝑣𝑚𝑤subscript𝑣0𝑚𝑑subscript𝑣0v\leq m(w)\!\cdot\!v_{0}=m(d)\!\cdot\!v_{0}italic_v ≤ italic_m ( italic_w ) ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_m ( italic_d ) ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, i.e.,

BvKB(m(d)v0)K¯=BdB¯K=BdP¯.𝐵𝑣𝐾¯𝐵𝑚𝑑subscript𝑣0𝐾¯𝐵𝑑𝐵𝐾¯𝐵𝑑𝑃BvK\subset\overline{B(m(d)\!\cdot\!v_{0})K}=\overline{BdB}\;K=\overline{BdP}.italic_B italic_v italic_K ⊂ ¯ start_ARG italic_B ( italic_m ( italic_d ) ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) italic_K end_ARG = ¯ start_ARG italic_B italic_d italic_B end_ARG italic_K = ¯ start_ARG italic_B italic_d italic_P end_ARG .

Since BvKBν(v)P𝐵𝑣𝐾𝐵𝜈𝑣𝑃BvK\subset B\nu(v)Pitalic_B italic_v italic_K ⊂ italic_B italic_ν ( italic_v ) italic_P, we have Bν(v)PBdP¯𝐵𝜈𝑣𝑃¯𝐵𝑑𝑃B\nu(v)P\cap\overline{BdP}\neq\emptysetitalic_B italic_ν ( italic_v ) italic_P ∩ ¯ start_ARG italic_B italic_d italic_P end_ARG ≠ ∅. Thus ν(v)dw𝜈𝑣𝑑𝑤\nu(v)\leq d\leq witalic_ν ( italic_v ) ≤ italic_d ≤ italic_w. This shows that ν(v)𝜈𝑣\nu(v)italic_ν ( italic_v ) is the unique minimal element in Wv0(v)subscript𝑊subscript𝑣0𝑣W_{v_{0}}(v)italic_W start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ).

Q.E.D.

Thus, our Theorem 2.2 generalizes [13, Theorem 5.2.5].

9.4. An example where Zv0(v)Yv0(v)subscript𝑍subscript𝑣0𝑣subscript𝑌subscript𝑣0𝑣Z_{v_{0}}(v)\neq Y_{v_{0}}(v)italic_Z start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) ≠ italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v )

Let G=SL(4,)𝐺𝑆𝐿4G=SL(4,{\mathbb{C}})italic_G = italic_S italic_L ( 4 , blackboard_C ) and let θAut(G)𝜃Aut𝐺\theta\in{\rm Aut}(G)italic_θ ∈ roman_Aut ( italic_G ) be given by θ(g)=I2,2(gt)-1I2,2𝜃𝑔subscript𝐼22superscriptsuperscript𝑔𝑡1subscript𝐼22\theta(g)=I_{2,2}(g^{t})^{-1}I_{2,2}italic_θ ( italic_g ) = italic_I start_POSTSUBSCRIPT 2 , 2 end_POSTSUBSCRIPT ( italic_g start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_I start_POSTSUBSCRIPT 2 , 2 end_POSTSUBSCRIPT, where for gG𝑔𝐺g\in Gitalic_g ∈ italic_G, gtsuperscript𝑔𝑡g^{t}italic_g start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT denotes the transpose of g𝑔gitalic_g, and I2,2=diag(I2,-I2)subscript𝐼22diagsubscript𝐼2subscript𝐼2I_{2,2}={\rm diag}(I_{2},-I_{2})italic_I start_POSTSUBSCRIPT 2 , 2 end_POSTSUBSCRIPT = roman_diag ( italic_I start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , - italic_I start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) with I2subscript𝐼2I_{2}italic_I start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT being the 2×2222\times 22 × 2 identity matrix. Then K=S(GL(2,)×GL(2,))𝐾𝑆𝐺𝐿2𝐺𝐿2K=S(GL(2,{\mathbb{C}})\times GL(2,{\mathbb{C}}))italic_K = italic_S ( italic_G italic_L ( 2 , blackboard_C ) × italic_G italic_L ( 2 , blackboard_C ) ). Using the “kgb” command in the Atlas of Lie groups (www.liegroups.org) for the real form SU(2,2)𝑆𝑈22SU(2,2)italic_S italic_U ( 2 , 2 ) of G𝐺Gitalic_G, one knows that there are 6666 elements in V0subscript𝑉0V_{0}italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT. Take v0V0subscript𝑣0subscript𝑉0v_{0}\in V_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT to be the orbit labeled by 3333 in the Atlas and let v𝑣vitalic_v be the orbit labeled by 19191919. With respect to v0subscript𝑣0v_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, the simple roots α1subscript𝛼1\alpha_{1}italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and α3subscript𝛼3\alpha_{3}italic_α start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT are noncompact, while α2subscript𝛼2\alpha_{2}italic_α start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT is compact, where we use the Bourbaki labeling of roots. Then v=m(s3s2s1)v0M(W,S)v0𝑣𝑚subscript𝑠3subscript𝑠2subscript𝑠1subscript𝑣0𝑀𝑊𝑆subscript𝑣0v=m(s_{3}s_{2}s_{1})\!\cdot\!v_{0}\in M(W,S)\!\cdot\!v_{0}italic_v = italic_m ( italic_s start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_s start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ italic_M ( italic_W , italic_S ) ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT. By Example 8.11, l(z)=3𝑙𝑧3l(z)=3italic_l ( italic_z ) = 3 for every zZv0(v)𝑧subscript𝑍subscript𝑣0𝑣z\in Z_{v_{0}}(v)italic_z ∈ italic_Z start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ).

On the other hand, let y=s2s1s2s3W𝑦subscript𝑠2subscript𝑠1subscript𝑠2subscript𝑠3𝑊y=s_{2}s_{1}s_{2}s_{3}\in Witalic_y = italic_s start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_s start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_s start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ∈ italic_W. Then m(y)v0=vmax𝑚𝑦subscript𝑣0subscript𝑣maxm(y)\!\cdot\!v_{0}=v_{{\rm max}}italic_m ( italic_y ) ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_v start_POSTSUBSCRIPT roman_max end_POSTSUBSCRIPT, where vmaxsubscript𝑣maxv_{{\rm max}}italic_v start_POSTSUBSCRIPT roman_max end_POSTSUBSCRIPT is maximal element in V𝑉Vitalic_V. Since vvmax𝑣subscript𝑣maxv\leq v_{{\rm max}}italic_v ≤ italic_v start_POSTSUBSCRIPT roman_max end_POSTSUBSCRIPT, yWv0(v)𝑦subscript𝑊subscript𝑣0𝑣y\in W_{v_{0}}(v)italic_y ∈ italic_W start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ). We claim that yYv0(v)𝑦subscript𝑌subscript𝑣0𝑣y\in Y_{v_{0}}(v)italic_y ∈ italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ). Indeed, if yYv0(v)𝑦subscript𝑌subscript𝑣0𝑣y\notin Y_{v_{0}}(v)italic_y ∉ italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ), then there exists yYv0(v)superscript𝑦subscript𝑌subscript𝑣0𝑣y^{\prime}\in Y_{v_{0}}(v)italic_y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) with y<ysuperscript𝑦𝑦y^{\prime}<yitalic_y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT < italic_y. Since l(y)l(v)=3𝑙𝑦𝑙𝑣3l(y)\geq l(v)=3italic_l ( italic_y ) ≥ italic_l ( italic_v ) = 3, we must have l(y)=3𝑙𝑦3l(y)=3italic_l ( italic_y ) = 3. Now there are exactly three subwords ysuperscript𝑦y^{\prime}italic_y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT of the word y=s2s1s2s3𝑦subscript𝑠2subscript𝑠1subscript𝑠2subscript𝑠3y=s_{2}s_{1}s_{2}s_{3}italic_y = italic_s start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_s start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_s start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT with length 3333, namely, y=s1s2s3superscript𝑦subscript𝑠1subscript𝑠2subscript𝑠3y^{\prime}=s_{1}s_{2}s_{3}italic_y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_s start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_s start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT or s2s1s3subscript𝑠2subscript𝑠1subscript𝑠3s_{2}s_{1}s_{3}italic_s start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_s start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT, or s2s1s2subscript𝑠2subscript𝑠1subscript𝑠2s_{2}s_{1}s_{2}italic_s start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_s start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT, and one checks directly that neither of these three choices gives vm(y)v0𝑣𝑚superscript𝑦subscript𝑣0v\leq m(y^{\prime})\!\cdot\!v_{0}italic_v ≤ italic_m ( italic_y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ⋅ italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT. Thus yYv0(v)𝑦subscript𝑌subscript𝑣0𝑣y\in Y_{v_{0}}(v)italic_y ∈ italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ). Since l(y)=4𝑙𝑦4l(y)=4italic_l ( italic_y ) = 4, yZv0(v)𝑦subscript𝑍subscript𝑣0𝑣y\notin Z_{v_{0}}(v)italic_y ∉ italic_Z start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ). We thus have an example in which Zv0(v)subscript𝑍subscript𝑣0𝑣Z_{v_{0}}(v)italic_Z start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ) is a proper subset of Yv0(v)subscript𝑌subscript𝑣0𝑣Y_{v_{0}}(v)italic_Y start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v ).

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